• # Paper 1, Section I, F

State and prove the Bolzano-Weierstrass theorem.

Consider a bounded sequence $\left(x_{n}\right)$. Prove that if every convergent subsequence of $\left(x_{n}\right)$ converges to the same limit $L$ then $\left(x_{n}\right)$ converges to $L$.

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• # Paper 1, Section I, F

State and prove the alternating series test. Hence show that the series $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}$ converges. Show also that

$\frac{7}{12} \leqslant \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} \leqslant \frac{47}{60}$

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• # Paper 1, Section II, F

(a) Let $\sum_{n=0}^{\infty} a_{n} z^{n}$ be a power series with $a_{n} \in \mathbb{C}$. Show that there exists $R \in[0, \infty]$ (called the radius of convergence) such that the series is absolutely convergent when $|z| but is divergent when $|z|>R$.

Suppose that the radius of convergence of the series $\sum_{n=0}^{\infty} a_{n} z^{n}$ is $R=2$. For a fixed positive integer $k$, find the radii of convergence of the following series. [You may assume that $\lim _{n \rightarrow \infty}\left|a_{n}\right|^{1 / n}$ exists.] (i) $\sum_{n=0}^{\infty} a_{n}^{k} z^{n}$. (ii) $\sum_{n=0}^{\infty} a_{n} z^{k n}$. (iii) $\sum_{n=0}^{\infty} a_{n} z^{n^{2}}$.

(b) Suppose that there exist values of $z$ for which $\sum_{n=0}^{\infty} b_{n} e^{n z}$ converges and values for which it diverges. Show that there exists a real number $S$ such that $\sum_{n=0}^{\infty} b_{n} e^{n z}$ diverges whenever $\operatorname{Re}(z)>S$ and converges whenever $\operatorname{Re}(z).

Determine the set of values of $z$ for which

$\sum_{n=0}^{\infty} \frac{2^{n} e^{i n z}}{(n+1)^{2}}$

converges.

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• # Paper 1, Section II, F

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be $n$-times differentiable, for some $n>0$.

(a) State and prove Taylor's theorem for $f$, with the Lagrange form of the remainder. [You may assume Rolle's theorem.]

(b) Suppose that $f: \mathbb{R} \rightarrow \mathbb{R}$ is an infinitely differentiable function such that $f(0)=1$ and $f^{\prime}(0)=0$, and satisfying the differential equation $f^{\prime \prime}(x)=-f(x)$. Prove carefully that

$f(x)=\sum_{k=0}^{\infty}(-1)^{k} \frac{x^{2 k}}{(2 k) !}$

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• # Paper 1, Section II, F

Let $f:[a, b] \rightarrow \mathbb{R}$ be a continuous function.

(a) Let $m=\min _{x \in[a, b]} f(x)$ and $M=\max _{x \in[a, b]} f(x)$. If $g:[a, b] \rightarrow \mathbb{R}$ is a positive continuous function, prove that

$m \int_{a}^{b} g(x) d x \leqslant \int_{a}^{b} f(x) g(x) d x \leqslant M \int_{a}^{b} g(x) d x$

directly from the definition of the Riemann integral.

(b) Let $f:[0,1] \rightarrow \mathbb{R}$ be a continuous function. Show that

$\int_{0}^{1 / \sqrt{n}} n f(x) e^{-n x} d x \rightarrow f(0)$

as $n \rightarrow \infty$, and deduce that

$\int_{0}^{1} n f(x) e^{-n x} d x \rightarrow f(0)$

as $n \rightarrow \infty$

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• # Paper 1, Section II, F

(a) State the intermediate value theorem. Show that if $f: \mathbb{R} \rightarrow \mathbb{R}$ is a continuous bijection and $x_{1} then either $f\left(x_{1}\right) or $f\left(x_{1}\right)>f\left(x_{2}\right)>f\left(x_{3}\right)$. Deduce that $f$ is either strictly increasing or strictly decreasing.

(b) Let $f: \mathbb{R} \rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ be functions. Which of the following statements are true, and which can be false? Give a proof or counterexample as appropriate.

(i) If $f$ and $g$ are continuous then $f \circ g$ is continuous.

(ii) If $g$ is strictly increasing and $f \circ g$ is continuous then $f$ is continuous.

(iii) If $f$ is continuous and a bijection then $f^{-1}$ is continuous.

(iv) If $f$ is differentiable and a bijection then $f^{-1}$ is differentiable.

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• # Paper 1, Section I, E

(a) Let $f$ be continuous in $[a, b]$, and let $g$ be strictly monotonic in $[\alpha, \beta]$, with a continuous derivative there, and suppose that $a=g(\alpha)$ and $b=g(\beta)$. Prove that

$\int_{a}^{b} f(x) d x=\int_{\alpha}^{\beta} f(g(u)) g^{\prime}(u) d u$

[Any version of the fundamental theorem of calculus may be used providing it is quoted correctly.]

(b) Justifying carefully the steps in your argument, show that the improper Riemann integral

$\int_{0}^{e^{-1}} \frac{d x}{x\left(\log \frac{1}{x}\right)^{\theta}}$

converges for $\theta>1$, and evaluate it.

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• # Paper 1, Section II, D

(a) State Rolle's theorem. Show that if $f: \mathbb{R} \rightarrow \mathbb{R}$ is $N+1$ times differentiable and $x \in \mathbb{R}$ then

$f(x)=f(0)+f^{\prime}(0) x+\frac{f^{\prime \prime}(0)}{2 !} x^{2}+\ldots+\frac{f^{(N)}(0)}{N !} x^{N}+\frac{f^{(N+1)}(\theta x)}{(N+1) !} x^{N+1}$

for some $0<\theta<1$. Hence, or otherwise, show that if $f^{\prime}(x)=0$ for all $x \in \mathbb{R}$ then $f$ is constant.

(b) Let $s: \mathbb{R} \rightarrow \mathbb{R}$ and $c: \mathbb{R} \rightarrow \mathbb{R}$ be differentiable functions such that

$s^{\prime}(x)=c(x), \quad c^{\prime}(x)=-s(x), \quad s(0)=0 \quad \text { and } \quad c(0)=1$

Prove that (i) $s(x) c(a-x)+c(x) s(a-x)$ is independent of $x$, (ii) $s(x+y)=s(x) c(y)+c(x) s(y)$, (iii) $s(x)^{2}+c(x)^{2}=1$.

Show that $c(1)>0$ and $c(2)<0$. Deduce there exists $1 such that $s(2 k)=c(k)=0$ and $s(x+4 k)=s(x)$.

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• # Paper 1, Section II, F

(a) Let $\left(x_{n}\right)$ be a bounded sequence of real numbers. Show that $\left(x_{n}\right)$ has a convergent subsequence.

(b) Let $\left(z_{n}\right)$ be a bounded sequence of complex numbers. For each $n \geqslant 1$, write $z_{n}=x_{n}+i y_{n}$. Show that $\left(z_{n}\right)$ has a subsequence $\left(z_{n_{j}}\right)$ such that $\left(x_{n_{j}}\right)$ converges. Hence, or otherwise, show that $\left(z_{n}\right)$ has a convergent subsequence.

(c) Write $\mathbb{N}=\{1,2,3, \ldots\}$ for the set of positive integers. Let $M$ be a positive real number, and for each $i \in \mathbb{N}$, let $X^{(i)}=\left(x_{1}^{(i)}, x_{2}^{(i)}, x_{3}^{(i)}, \ldots\right)$ be a sequence of real numbers with $\left|x_{j}^{(i)}\right| \leqslant M$ for all $i, j \in \mathbb{N}$. By induction on $i$ or otherwise, show that there exist sequences $N^{(i)}=\left(n_{1}^{(i)}, n_{2}^{(i)}, n_{3}^{(i)}, \ldots\right)$ of positive integers with the following properties:

• for all $i \in \mathbb{N}$, the sequence $N^{(i)}$ is strictly increasing;

• for all $i \in \mathbb{N}, N^{(i+1)}$ is a subsequence of $N^{(i)} ;$ and

• for all $k \in \mathbb{N}$ and all $i \in \mathbb{N}$ with $1 \leqslant i \leqslant k$, the sequence

$\left(x_{n_{1}^{(k)}}^{(i)}, x_{n_{2}^{(k)}}^{(i)}, x_{n_{3}^{(k)}}^{(i)}, \ldots\right)$

converges.

Hence, or otherwise, show that there exists a strictly increasing sequence $\left(m_{j}\right)$ of positive integers such that for all $i \in \mathbb{N}$ the sequence $\left(x_{m_{1}}^{(i)}, x_{m_{2}}^{(i)}, x_{m_{3}}^{(i)}, \ldots\right)$ converges.

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• # Paper 1, Section I, E

State the Bolzano-Weierstrass theorem.

Let $\left(a_{n}\right)$ be a sequence of non-zero real numbers. Which of the following conditions is sufficient to ensure that $\left(1 / a_{n}\right)$ converges? Give a proof or counter-example as appropriate.

(i) $a_{n} \rightarrow \ell$ for some real number $\ell$.

(ii) $a_{n} \rightarrow \ell$ for some non-zero real number $\ell$.

(iii) $\left(a_{n}\right)$ has no convergent subsequence.

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• # Paper 1, Section I, F

Let $\sum_{n=1}^{\infty} a_{n} x^{n}$ be a real power series that diverges for at least one value of $x$. Show that there exists a non-negative real number $R$ such that $\sum_{n=1}^{\infty} a_{n} x^{n}$ converges absolutely whenever $|x| and diverges whenever $|x|>R$.

Find, with justification, such a number $R$ for each of the following real power series:

(i) $\sum_{n=1}^{\infty} \frac{x^{n}}{3^{n}}$;

(ii) $\sum_{n=1}^{\infty} x^{n}\left(1+\frac{1}{n}\right)^{n}$.

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• # Paper 1, Section II, D

State and prove the Intermediate Value Theorem.

State the Mean Value Theorem.

Suppose that the function $g$ is differentiable everywhere in some open interval containing $[a, b]$, and that $g^{\prime}(a). By considering the functions $h$ and $f$ defined by

$h(x)=\frac{g(x)-g(a)}{x-a}(a

and

$f(x)=\frac{g(b)-g(x)}{b-x}(a \leqslant x

or otherwise, show that there is a subinterval $[\alpha, \beta] \subseteq[a, b]$ such that

$\frac{g(\beta)-g(\alpha)}{\beta-\alpha}=k$

Deduce that there exists $c \in(a, b)$ with $g^{\prime}(c)=k$.

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• # Paper 1, Section II, D

Let $g: \mathbb{R} \rightarrow \mathbb{R}$ be a function that is continuous at at least one point $z \in \mathbb{R}$. Suppose further that $g$ satisfies

$g(x+y)=g(x)+g(y)$

for all $x, y \in \mathbb{R}$. Show that $g$ is continuous on $\mathbb{R}$.

Show that there exists a constant $c$ such that $g(x)=c x$ for all $x \in \mathbb{R}$.

Suppose that $h: \mathbb{R} \rightarrow \mathbb{R}$ is a continuous function defined on $\mathbb{R}$ and that $h$ satisfies the equation

$h(x+y)=h(x) h(y)$

for all $x, y \in \mathbb{R}$. Show that $h$ is either identically zero or everywhere positive. What is the general form for $h$ ?

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• # Paper 1, Section II, E

Let $\left(a_{n}\right)$ and $\left(b_{n}\right)$ be sequences of positive real numbers. Let $s_{n}=\sum_{i=1}^{n} a_{i}$.

(a) Show that if $\sum a_{n}$ and $\sum b_{n}$ converge then so does $\sum\left(a_{n}^{2}+b_{n}^{2}\right)^{1 / 2}$.

(b) Show that if $\sum a_{n}$ converges then $\sum \sqrt{a_{n} a_{n+1}}$ converges. Is the converse true?

(c) Show that if $\sum a_{n}$ diverges then $\sum \frac{a_{n}}{s_{n}}$ diverges. Is the converse true?

$[$ For part (c), it may help to show that for any $N \in \mathbb{N}$ there exist $m \geqslant n \geqslant N$ with

$\left.\frac{a_{n+1}}{s_{n+1}}+\frac{a_{n+2}}{s_{n+2}}+\ldots+\frac{a_{m}}{s_{m}} \geqslant \frac{1}{2} .\right]$

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• # Paper 1, Section II, F

Let $f:[0,1] \rightarrow \mathbb{R}$ be a bounded function. Define the upper and lower integrals of $f$. What does it mean to say that $f$ is Riemann integrable? If $f$ is Riemann integrable, what is the Riemann integral $\int_{0}^{1} f(x) d x$ ?

Which of the following functions $f:[0,1] \rightarrow \mathbb{R}$ are Riemann integrable? For those that are Riemann integrable, find $\int_{0}^{1} f(x) d x$. Justify your answers.

(i) $f(x)= \begin{cases}1 & \text { if } x \in \mathbb{Q} \\ 0 & \text { if } x \notin \mathbb{Q}\end{cases}$

(ii) $f(x)=\left\{\begin{array}{ll}1 & \text { if } x \in A \\ 0 & \text { if } x \notin A\end{array}\right.$,

where $A=\{x \in[0,1]: x$ has a base-3 expansion containing a 1$\}$;

[Hint: You may find it helpful to note, for example, that $\frac{2}{3} \in A$ as one of the base-3 expansions of $\frac{2}{3}$ is $\left.0.1222 \ldots .\right]$

(iii) $f(x)=\left\{\begin{array}{ll}1 & \text { if } x \in B \\ 0 & \text { if } x \notin B\end{array}\right.$,

where $B=\{x \in[0,1]: x$ has a base $-3$ expansion containing infinitely many $1 \mathrm{~s}\}$.

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• # Paper 1, Section I, $4 \mathrm{D}$

Define the radius of convergence $R$ of a complex power series $\sum a_{n} z^{n}$. Prove that $\sum a_{n} z^{n}$ converges whenever $|z| and diverges whenever $|z|>R$.

If $\left|a_{n}\right| \leqslant\left|b_{n}\right|$ for all $n$ does it follow that the radius of convergence of $\sum a_{n} z^{n}$ is at least that of $\sum b_{n} z^{n}$ ? Justify your answer.

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• # Paper 1, Section I, E

Prove that an increasing sequence in $\mathbb{R}$ that is bounded above converges.

Let $f: \mathbb{R} \rightarrow(0, \infty)$ be a decreasing function. Let $x_{1}=1$ and $x_{n+1}=x_{n}+f\left(x_{n}\right)$. Prove that $x_{n} \rightarrow \infty$ as $n \rightarrow \infty$.

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• # Paper 1, Section II, $10 F$

(a) Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be differentiable at $x_{0} \in \mathbb{R}$. Show that $f$ is continuous at $x_{0}$.

(b) State the Mean Value Theorem. Prove the following inequalities:

$\left|\cos \left(e^{-x}\right)-\cos \left(e^{-y}\right)\right| \leqslant|x-y| \quad \text { for } x, y \geqslant 0$

and

$\log (1+x) \leqslant \frac{x}{\sqrt{1+x}} \text { for } x \geqslant 0 .$

(c) Determine at which points the following functions from $\mathbb{R}$ to $\mathbb{R}$ are differentiable, and find their derivatives at the points at which they are differentiable:

$f(x)=\left\{\begin{array}{ll} |x|^{x} & \text { if } x \neq 0 \\ 1 & \text { if } x=0, \end{array} \quad g(x)=\cos (|x|), \quad h(x)=x|x|\right.$

(d) Determine the points at which the following function from $\mathbb{R}$ to $\mathbb{R}$ is continuous:

$f(x)= \begin{cases}0 & \text { if } x \notin \mathbb{Q} \text { or } x=0 \\ 1 / q & \text { if } x=p / q \text { where } p \in \mathbb{Z} \backslash\{0\} \text { and } q \in \mathbb{N} \text { are relatively prime. }\end{cases}$

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• # Paper 1, Section II, D

(a) Let $q_{1}, q_{2}, \ldots$ be a fixed enumeration of the rationals in $[0,1]$. For positive reals $a_{1}, a_{2}, \ldots$, define a function $f$ from $[0,1]$ to $\mathbb{R}$ by setting $f\left(q_{n}\right)=a_{n}$ for each $n$ and $f(x)=0$ for $x$ irrational. Prove that if $a_{n} \rightarrow 0$ then $f$ is Riemann integrable. If $a_{n} \rightarrow 0$, can $f$ be Riemann integrable? Justify your answer.

(b) State and prove the Fundamental Theorem of Calculus.

Let $f$ be a differentiable function from $\mathbb{R}$ to $\mathbb{R}$, and set $g(x)=f^{\prime}(x)$ for $0 \leqslant x \leqslant 1$. Must $g$ be Riemann integrable on $[0,1]$ ?

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• # Paper 1, Section II, E

State and prove the Comparison Test for real series.

Assume $0 \leqslant x_{n}<1$ for all $n \in \mathbb{N}$. Show that if $\sum x_{n}$ converges, then so do $\sum x_{n}^{2}$ and $\sum \frac{x_{n}}{1-x_{n}}$. In each case, does the converse hold? Justify your answers.

Let $\left(x_{n}\right)$ be a decreasing sequence of positive reals. Show that if $\sum x_{n}$ converges, then $n x_{n} \rightarrow 0$ as $n \rightarrow \infty$. Does the converse hold? If $\sum x_{n}$ converges, must it be the case that $(n \log n) x_{n} \rightarrow 0$ as $n \rightarrow \infty$ ? Justify your answers.

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• # Paper 1, Section II, F

(a) Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function, and let $x \in \mathbb{R}$. Define what it means for $f$ to be continuous at $x$. Show that $f$ is continuous at $x$ if and only if $f\left(x_{n}\right) \rightarrow f(x)$ for every sequence $\left(x_{n}\right)$ with $x_{n} \rightarrow x$.

(b) Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a non-constant polynomial. Show that its image $\{f(x): x \in \mathbb{R}\}$ is either the real line $\mathbb{R}$, the interval $[a, \infty)$ for some $a \in \mathbb{R}$, or the interval $(-\infty, a]$ for some $a \in \mathbb{R}$.

(c) Let $\alpha>1$, let $f:(0, \infty) \rightarrow \mathbb{R}$ be continuous, and assume that $f(x)=f\left(x^{\alpha}\right)$ holds for all $x>0$. Show that $f$ must be constant.

Is this also true when the condition that $f$ be continuous is dropped?

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• # Paper 1, Section I, $4 \mathrm{E}$

Show that if the power series $\sum_{n=0}^{\infty} a_{n} z^{n}(z \in \mathbb{C})$ converges for some fixed $z=z_{0}$, then it converges absolutely for every $z$ satisfying $|z|<\left|z_{0}\right|$.

Define the radius of convergence of a power series.

Give an example of $v \in \mathbb{C}$ and an example of $w \in \mathbb{C}$ such that $|v|=|w|=1, \sum_{n=1}^{\infty} \frac{v^{n}}{n}$ converges and $\sum_{n=1}^{\infty} \frac{w^{n}}{n}$ diverges. [You may assume results about standard series without proof.] Use this to find the radius of convergence of the power series $\sum_{n=1}^{\infty} \frac{z^{n}}{n}$.

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• # Paper 1, Section I, F

Given an increasing sequence of non-negative real numbers $\left(a_{n}\right)_{n=1}^{\infty}$, let

$s_{n}=\frac{1}{n} \sum_{k=1}^{n} a_{k}$

Prove that if $s_{n} \rightarrow x$ as $n \rightarrow \infty$ for some $x \in \mathbb{R}$ then also $a_{n} \rightarrow x$ as $n \rightarrow \infty$

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• # Paper 1, Section II, D

Let $a, b \in \mathbb{R}$ with $a and let $f:(a, b) \rightarrow \mathbb{R}$.

(a) Define what it means for $f$ to be continuous at $y_{0} \in(a, b)$.

$f$ is said to have a local minimum at $c \in(a, b)$ if there is some $\varepsilon>0$ such that $f(c) \leqslant f(x)$ whenever $x \in(a, b)$ and $|x-c|<\varepsilon$.

If $f$ has a local minimum at $c \in(a, b)$ and $f$ is differentiable at $c$, show that $f^{\prime}(c)=0$.

(b) $f$ is said to be convex if

$f(\lambda x+(1-\lambda) y) \leqslant \lambda f(x)+(1-\lambda) f(y)$

for every $x, y \in(a, b)$ and $\lambda \in[0,1]$. If $f$ is convex, $r \in \mathbb{R}$ and $\left[y_{0}-|r|, y_{0}+|r|\right] \subset(a, b)$, prove that

$(1+\lambda) f\left(y_{0}\right)-\lambda f\left(y_{0}-r\right) \leqslant f\left(y_{0}+\lambda r\right) \leqslant(1-\lambda) f\left(y_{0}\right)+\lambda f\left(y_{0}+r\right)$

for every $\lambda \in[0,1]$.

Deduce that if $f$ is convex then $f$ is continuous.

If $f$ is convex and has a local minimum at $c \in(a, b)$, prove that $f$ has a global minimum at $c$, i.e., that $f(x) \geqslant f(c)$ for every $x \in(a, b)$. [Hint: argue by contradiction.] Must $f$ be differentiable at $c$ ? Justify your answer.

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• # Paper 1, Section II, D

(a) State the Intermediate Value Theorem.

(b) Define what it means for a function $f: \mathbb{R} \rightarrow \mathbb{R}$ to be differentiable at a point $a \in \mathbb{R}$. If $f$ is differentiable everywhere on $\mathbb{R}$, must $f^{\prime}$ be continuous everywhere? Justify your answer.

State the Mean Value Theorem.

(c) Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be differentiable everywhere. Let $a, b \in \mathbb{R}$ with $a.

If $f^{\prime}(a) \leqslant y \leqslant f^{\prime}(b)$, prove that there exists $c \in[a, b]$ such that $f^{\prime}(c)=y$. [Hint: consider the function $g$ defined by

$g(x)=\frac{f(x)-f(a)}{x-a}$

if $x \neq a$ and $\left.g(a)=f^{\prime}(a) .\right]$

If additionally $f(a) \leqslant 0 \leqslant f(b)$, deduce that there exists $d \in[a, b]$ such that $f^{\prime}(d)+f(d)=y$.

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• # Paper 1, Section II, E

Let $f:[a, b] \rightarrow \mathbb{R}$ be a bounded function defined on the closed, bounded interval $[a, b]$ of $\mathbb{R}$. Suppose that for every $\varepsilon>0$ there is a dissection $\mathcal{D}$ of $[a, b]$ such that $S_{\mathcal{D}}(f)-s_{\mathcal{D}}(f)<\varepsilon$, where $s_{\mathcal{D}}(f)$ and $S_{\mathcal{D}}(f)$ denote the lower and upper Riemann sums of $f$ for the dissection $\mathcal{D}$. Deduce that $f$ is Riemann integrable. [You may assume without proof that $s_{\mathcal{D}}(f) \leqslant S_{\mathcal{D}^{\prime}}(f)$ for all dissections $\mathcal{D}$ and $\mathcal{D}^{\prime}$ of $\left.[a, b] .\right]$

Prove that if $f:[a, b] \rightarrow \mathbb{R}$ is continuous, then $f$ is Riemann integrable.

Let $g:(0,1] \rightarrow \mathbb{R}$ be a bounded continuous function. Show that for any $\lambda \in \mathbb{R}$, the function $f:[0,1] \rightarrow \mathbb{R}$ defined by

$f(x)= \begin{cases}g(x) & \text { if } 0

is Riemann integrable.

Let $f:[a, b] \rightarrow \mathbb{R}$ be a differentiable function with one-sided derivatives at the endpoints. Suppose that the derivative $f^{\prime}$ is (bounded and) Riemann integrable. Show that

$\int_{a}^{b} f^{\prime}(x) d x=f(b)-f(a)$

[You may use the Mean Value Theorem without proof.]

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• # Paper 1, Section II, F

(a) Let $\left(x_{n}\right)_{n=1}^{\infty}$ be a non-negative and decreasing sequence of real numbers. Prove that $\sum_{n=1}^{\infty} x_{n}$ converges if and only if $\sum_{k=0}^{\infty} 2^{k} x_{2^{k}}$ converges.

(b) For $s \in \mathbb{R}$, prove that $\sum_{n=1}^{\infty} n^{-s}$ converges if and only if $s>1$.

(c) For any $k \in \mathbb{N}$, prove that

$\lim _{n \rightarrow \infty} 2^{-n} n^{k}=0$

(d) The sequence $\left(a_{n}\right)_{n=0}^{\infty}$ is defined by $a_{0}=1$ and $a_{n+1}=2^{a_{n}}$ for $n \geqslant 0$. For any $k \in \mathbb{N}$, prove that

$\lim _{n \rightarrow \infty} \frac{2^{n^{k}}}{a_{n}}=0$

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• # Paper 1, Section I, D

What does it mean to say that a sequence of real numbers $\left(x_{n}\right)$ converges to $x$ ? Suppose that $\left(x_{n}\right)$ converges to $x$. Show that the sequence $\left(y_{n}\right)$ given by

$y_{n}=\frac{1}{n} \sum_{i=1}^{n} x_{i}$

also converges to $x$.

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• # Paper 1, Section I, F

Let $a_{n}$ be the number of pairs of integers $(x, y) \in \mathbb{Z}^{2}$ such that $x^{2}+y^{2} \leqslant n^{2}$. What is the radius of convergence of the series $\sum_{n=0}^{\infty} a_{n} z^{n}$ ? [You may use the comparison test, provided you state it clearly.]

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• # Paper 1, Section II, 12F

Let $f:[0,1] \rightarrow \mathbb{R}$ satisfy $|f(x)-f(y)| \leqslant|x-y|$ for all $x, y \in[0,1]$.

Show that $f$ is continuous and that for all $\varepsilon>0$, there exists a piecewise constant function $g$ such that

$\sup _{x \in[0,1]}|f(x)-g(x)| \leqslant \varepsilon .$

For all integers $n \geqslant 1$, let $u_{n}=\int_{0}^{1} f(t) \cos (n t) d t$. Show that the sequence $\left(u_{n}\right)$ converges to 0 .

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• # Paper 1, Section II, D

If $\left(x_{n}\right)$ and $\left(y_{n}\right)$ are sequences converging to $x$ and $y$ respectively, show that the sequence $\left(x_{n}+y_{n}\right)$ converges to $x+y$.

If $x_{n} \neq 0$ for all $n$ and $x \neq 0$, show that the sequence $\left(\frac{1}{x_{n}}\right)$ converges to $\frac{1}{x}$.

(a) Find $\lim _{n \rightarrow \infty}\left(\sqrt{n^{2}+n}-n\right)$.

(b) Determine whether $\sum_{n=1}^{\infty} \frac{\sqrt{n+1}-\sqrt{n}}{\sqrt{n}}$ converges.

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• # Paper 1, Section II, E

Let $f: \mathbb{R} \rightarrow \mathbb{R}$. We say that $x \in \mathbb{R}$ is a real root of $f$ if $f(x)=0$. Show that if $f$ is differentiable and has $k$ distinct real roots, then $f^{\prime}$ has at least $k-1$ real roots. [Rolle's theorem may be used, provided you state it clearly.]

Let $p(x)=\sum_{i=1}^{n} a_{i} x^{d_{i}}$ be a polynomial in $x$, where all $a_{i} \neq 0$ and $d_{i+1}>d_{i}$. (In other words, the $a_{i}$ are the nonzero coefficients of the polynomial, arranged in order of increasing power of $x$.) The number of sign changes in the coefficients of $p$ is the number of $i$ for which $a_{i} a_{i+1}<0$. For example, the polynomial $x^{5}-x^{3}-x^{2}+1$ has 2 sign changes. Show by induction on $n$ that the number of positive real roots of $p$ is less than or equal to the number of sign changes in its coefficients.

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• # Paper 1, Section II, E

State the Bolzano-Weierstrass theorem. Use it to show that a continuous function $f:[a, b] \rightarrow \mathbb{R}$ attains a global maximum; that is, there is a real number $c \in[a, b]$ such that $f(c) \geqslant f(x)$ for all $x \in[a, b]$.

A function $f$ is said to attain a local maximum at $c \in \mathbb{R}$ if there is some $\varepsilon>0$ such that $f(c) \geqslant f(x)$ whenever $|x-c|<\varepsilon$. Suppose that $f: \mathbb{R} \rightarrow \mathbb{R}$ is twice differentiable, and that $f^{\prime \prime}(x)<0$ for all $x \in \mathbb{R}$. Show that there is at most one $c \in \mathbb{R}$ at which $f$ attains a local maximum.

If there is a constant $K<0$ such that $f^{\prime \prime}(x) for all $x \in \mathbb{R}$, show that $f$ attains a global maximum. [Hint: if $g^{\prime}(x)<0$ for all $x \in \mathbb{R}$, then $g$ is decreasing.]

Must $f: \mathbb{R} \rightarrow \mathbb{R}$ attain a global maximum if we merely require $f^{\prime \prime}(x)<0$ for all $x \in \mathbb{R} ?$ Justify your answer.

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• # Paper 1, Section I, $3 \mathbf{F}$

Find the following limits: (a) $\lim _{x \rightarrow 0} \frac{\sin x}{x}$ (b) $\lim _{x \rightarrow 0}(1+x)^{1 / x}$ (c) $\lim _{x \rightarrow \infty} \frac{(1+x)^{\frac{x}{1+x}} \cos ^{4} x}{e^{x}}$

[You may use standard results provided that they are clearly stated.]

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• # Paper 1, Section I, E

Let $\sum_{n \geqslant 0} a_{n} z^{n}$ be a complex power series. State carefully what it means for the power series to have radius of convergence $R$, with $0 \leqslant R \leqslant \infty$.

Find the radius of convergence of $\sum_{n \geqslant 0} p(n) z^{n}$, where $p(n)$ is a fixed polynomial in $n$ with coefficients in $\mathbb{C}$.

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• # Paper 1, Section II, $11 D$

(i) State and prove the intermediate value theorem.

(ii) Let $f:[0,1] \rightarrow \mathbb{R}$ be a continuous function. The chord joining the points $(\alpha, f(\alpha))$ and $(\beta, f(\beta))$ of the curve $y=f(x)$ is said to be horizontal if $f(\alpha)=f(\beta)$. Suppose that the chord joining the points $(0, f(0))$ and $(1, f(1))$ is horizontal. By considering the function $g$ defined on $\left[0, \frac{1}{2}\right]$ by

$g(x)=f\left(x+\frac{1}{2}\right)-f(x)$

or otherwise, show that the curve $y=f(x)$ has a horizontal chord of length $\frac{1}{2}$ in $[0,1]$. Show, more generally, that it has a horizontal chord of length $\frac{1}{n}$ for each positive integer $n$.

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• # Paper 1, Section II, 10D

(a) For real numbers $a, b$ such that $a, let $f:[a, b] \rightarrow \mathbb{R}$ be a continuous function. Prove that $f$ is bounded on $[a, b]$, and that $f$ attains its supremum and infimum on $[a, b]$.

(b) For $x \in \mathbb{R}$, define

$g(x)=\left\{\begin{array}{ll} |x|^{\frac{1}{2}} \sin (1 / \sin x), & x \neq n \pi \\ 0, & x=n \pi \end{array} \quad(n \in \mathbb{Z})\right.$

Find the set of points $x \in \mathbb{R}$ at which $g(x)$ is continuous.

Does $g$ attain its supremum on $[0, \pi] ?$

Does $g$ attain its supremum on $[\pi, 3 \pi / 2]$ ?

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• # Paper 1, Section II, E

Let $f:[0,1] \rightarrow \mathbb{R}$ be a bounded function, and let $\mathcal{D}_{n}$ denote the dissection $0<\frac{1}{n}<\frac{2}{n}<\cdots<\frac{n-1}{n}<1$ of $[0,1]$. Prove that $f$ is Riemann integrable if and only if the difference between the upper and lower sums of $f$ with respect to the dissection $\mathcal{D}_{n}$ tends to zero as $n$ tends to infinity.

Suppose that $f$ is Riemann integrable and $g: \mathbb{R} \rightarrow \mathbb{R}$ is continuously differentiable. Prove that $g \circ f$ is Riemann integrable.

[You may use the mean value theorem provided that it is clearly stated.]

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• # Paper 1, Section II, F

Let $\left(a_{n}\right),\left(b_{n}\right)$ be sequences of real numbers. Let $S_{n}=\sum_{j=1}^{n} a_{j}$ and set $S_{0}=0$. Show that for any $1 \leqslant m \leqslant n$ we have

$\sum_{j=m}^{n} a_{j} b_{j}=S_{n} b_{n}-S_{m-1} b_{m}+\sum_{j=m}^{n-1} S_{j}\left(b_{j}-b_{j+1}\right)$

Suppose that the series $\sum_{n \geqslant 1} a_{n}$ converges and that $\left(b_{n}\right)$ is bounded and monotonic. Does $\sum_{n \geqslant 1} a_{n} b_{n}$ converge?

Assume again that $\sum_{n \geqslant 1} a_{n}$ converges. Does $\sum_{n \geqslant 1} n^{1 / n} a_{n}$ converge?

[You may use the fact that a sequence of real numbers converges if and only if it is a Cauchy sequence.]

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• # Paper 1, Section I, $4 \mathbf{F}$

Find the radius of convergence of the following power series: (i) $\sum_{n \geqslant 1} \frac{n !}{n^{n}} z^{n}$; (ii) $\sum_{n \geqslant 1} n^{n} z^{n !}$.

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• # Paper 1, Section I, D

Show that every sequence of real numbers contains a monotone subsequence.

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• # Paper 1, Section II, D

(a) Show that for all $x \in \mathbb{R}$,

$\lim _{k \rightarrow \infty} 3^{k} \sin \left(x / 3^{k}\right)=x,$

stating carefully what properties of sin you are using.

Show that the series $\sum_{n \geqslant 1} 2^{n} \sin \left(x / 3^{n}\right)$ converges absolutely for all $x \in \mathbb{R}$.

(b) Let $\left(a_{n}\right)_{n \in \mathbb{N}}$ be a decreasing sequence of positive real numbers tending to zero. Show that for $\theta \in \mathbb{R}, \theta$ not a multiple of $2 \pi$, the series

$\sum_{n \geqslant 1} a_{n} e^{i n \theta}$

converges.

Hence, or otherwise, show that $\sum_{n \geqslant 1} \frac{\sin (n \theta)}{n}$ converges for all $\theta \in \mathbb{R}$.

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• # Paper 1, Section II, E

(i) Prove Taylor's Theorem for a function $f: \mathbb{R} \rightarrow \mathbb{R}$ differentiable $n$ times, in the following form: for every $x \in \mathbb{R}$ there exists $\theta$ with $0<\theta<1$ such that

$f(x)=\sum_{k=0}^{n-1} \frac{f^{(k)}(0)}{k !} x^{k}+\frac{f^{(n)}(\theta x)}{n !} x^{n}$

[You may assume Rolle's Theorem and the Mean Value Theorem; other results should be proved.]

(ii) The function $f: \mathbb{R} \rightarrow \mathbb{R}$ is twice differentiable, and satisfies the differential equation $f^{\prime \prime}-f=0$ with $f(0)=A, f^{\prime}(0)=B$. Show that $f$ is infinitely differentiable. Write down its Taylor series at the origin, and prove that it converges to $f$ at every point. Hence or otherwise show that for any $a, h \in \mathbb{R}$, the series

$\sum_{k=0}^{\infty} \frac{f^{(k)}(a)}{k !} h^{k}$

converges to $f(a+h)$.

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• # Paper 1, Section II, E

(i) State the Mean Value Theorem. Use it to show that if $f:(a, b) \rightarrow \mathbb{R}$ is a differentiable function whose derivative is identically zero, then $f$ is constant.

(ii) Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function and $\alpha>0$ a real number such that for all $x, y \in \mathbb{R}$,

$|f(x)-f(y)| \leqslant|x-y|^{\alpha} .$

Show that $f$ is continuous. Show moreover that if $\alpha>1$ then $f$ is constant.

(iii) Let $f:[a, b] \rightarrow \mathbb{R}$ be continuous, and differentiable on $(a, b)$. Assume also that the right derivative of $f$ at $a$ exists; that is, the limit

$\lim _{x \rightarrow a+} \frac{f(x)-f(a)}{x-a}$

exists. Show that for any $\epsilon>0$ there exists $x \in(a, b)$ satisfying

$\left|\frac{f(x)-f(a)}{x-a}-f^{\prime}(x)\right|<\epsilon .$

[You should not assume that $f^{\prime}$ is continuous.]

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• # Paper 1, Section II, F

Define what it means for a function $f:[0,1] \rightarrow \mathbb{R}$ to be (Riemann) integrable. Prove that $f$ is integrable whenever it is

(a) continuous,

(b) monotonic.

Let $\left\{q_{k}: k \in \mathbb{N}\right\}$ be an enumeration of all rational numbers in $[0,1)$. Define a function $f:[0,1] \rightarrow \mathbb{R}$ by $f(0)=0$,

$f(x)=\sum_{k \in Q(x)} 2^{-k}, \quad x \in(0,1]$

where

$Q(x)=\left\{k \in \mathbb{N}: q_{k} \in[0, x)\right\}$

Show that $f$ has a point of discontinuity in every interval $I \subset[0,1]$.

Is $f$ integrable? [Justify your answer.]

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• # Paper 1, Section I, D

Show that $\exp (x) \geqslant 1+x$ for $x \geqslant 0$.

Let $\left(a_{j}\right)$ be a sequence of positive real numbers. Show that for every $n$,

$\sum_{1}^{n} a_{j} \leqslant \prod_{1}^{n}\left(1+a_{j}\right) \leqslant \exp \left(\sum_{1}^{n} a_{j}\right)$

Deduce that $\prod_{1}^{n}\left(1+a_{j}\right)$ tends to a limit as $n \rightarrow \infty$ if and only if $\sum_{1}^{n} a_{j}$ does.

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• # Paper 1, Section I, F

(a) Suppose $b_{n} \geqslant b_{n+1} \geqslant 0$ for $n \geqslant 1$ and $b_{n} \rightarrow 0$. Show that $\sum_{n=1}^{\infty}(-1)^{n-1} b_{n}$ converges.

(b) Does the series $\sum_{n=2}^{\infty} \frac{1}{n \log n}$ converge or diverge? Explain your answer.

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• # Paper 1, Section II, D

(a) Determine the radius of convergence of each of the following power series:

$\sum_{n \geqslant 1} \frac{x^{n}}{n !}, \quad \sum_{n \geqslant 1} n ! x^{n}, \quad \sum_{n \geqslant 1}(n !)^{2} x^{n^{2}}$

(b) State Taylor's theorem.

Show that

$(1+x)^{1 / 2}=1+\sum_{n \geqslant 1} c_{n} x^{n}$

for all $x \in(0,1)$, where

$c_{n}=\frac{\frac{1}{2}\left(\frac{1}{2}-1\right) \ldots\left(\frac{1}{2}-n+1\right)}{n !}$

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• # Paper 1, Section II, E

(i) State (without proof) Rolle's Theorem.

(ii) State and prove the Mean Value Theorem.

(iii) Let $f, g:[a, b] \rightarrow \mathbb{R}$ be continuous, and differentiable on $(a, b)$ with $g^{\prime}(x) \neq 0$ for all $x \in(a, b)$. Show that there exists $\xi \in(a, b)$ such that

$\frac{f^{\prime}(\xi)}{g^{\prime}(\xi)}=\frac{f(b)-f(a)}{g(b)-g(a)}$

Deduce that if moreover $f(a)=g(a)=0$, and the limit

$\ell=\lim _{x \rightarrow a} \frac{f^{\prime}(x)}{g^{\prime}(x)}$

exists, then

$\frac{f(x)}{g(x)} \rightarrow \ell \text { as } x \rightarrow a$

(iv) Deduce that if $f: \mathbb{R} \rightarrow \mathbb{R}$ is twice differentiable then for any $a \in \mathbb{R}$

$f^{\prime \prime}(a)=\lim _{h \rightarrow 0} \frac{f(a+h)+f(a-h)-2 f(a)}{h^{2}} .$

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• # Paper 1, Section II, E

(a) Let $f:[a, b] \rightarrow \mathbb{R}$. Suppose that for every sequence $\left(x_{n}\right)$ in $[a, b]$ with limit $y \in[a, b]$, the sequence $\left(f\left(x_{n}\right)\right)$ converges to $f(y)$. Show that $f$ is continuous at $y$.

(b) State the Intermediate Value Theorem.

Let $f:[a, b] \rightarrow \mathbb{R}$ be a function with $f(a)=c. We say $f$ is injective if for all $x, y \in[a, b]$ with $x \neq y$, we have $f(x) \neq f(y)$. We say $f$ is strictly increasing if for all $x, y$ with $x, we have $f(x).

(i) Suppose $f$ is strictly increasing. Show that it is injective, and that if $f(x) then $x

(ii) Suppose $f$ is continuous and injective. Show that if $a then $c. Deduce that $f$ is strictly increasing.

(iii) Suppose $f$ is strictly increasing, and that for every $y \in[c, d]$ there exists $x \in[a, b]$ with $f(x)=y$. Show that $f$ is continuous at $b$. Deduce that $f$ is continuous on $[a, b]$.

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• # Paper 1, Section II, F

Fix a closed interval $[a, b]$. For a bounded function $f$ on $[a, b]$ and a dissection $\mathcal{D}$ of $[a, b]$, how are the lower sum $s(f, \mathcal{D})$ and upper sum $S(f, \mathcal{D})$ defined? Show that $s(f, \mathcal{D}) \leqslant S(f, \mathcal{D})$.

Suppose $\mathcal{D}^{\prime}$ is a dissection of $[a, b]$ such that $\mathcal{D} \subseteq \mathcal{D}^{\prime}$. Show that

$s(f, \mathcal{D}) \leqslant s\left(f, \mathcal{D}^{\prime}\right) \text { and } S\left(f, \mathcal{D}^{\prime}\right) \leqslant S(f, \mathcal{D})$

By using the above inequalities or otherwise, show that if $\mathcal{D}_{1}$ and $\mathcal{D}_{2}$ are two dissections of $[a, b]$ then

$s\left(f, \mathcal{D}_{1}\right) \leqslant S\left(f, \mathcal{D}_{2}\right)$

For a function $f$ and dissection $\mathcal{D}=\left\{x_{0}, \ldots, x_{n}\right\}$ let

$p(f, \mathcal{D})=\prod_{k=1}^{n}\left[1+\left(x_{k}-x_{k-1}\right) \inf _{x \in\left[x_{k-1}, x_{k}\right]} f(x)\right]$