Paper 1, Section II, D

Let $f$ be a continuous function from $(0,1)$ to $(0,1)$ such that $f(x)<x$ for every $0<x<1$. We write $f^{n}$ for the $n$-fold composition of $f$ with itself (so for example $\left.f^{2}(x)=f(f(x))\right)$.

(i) Prove that for every $0<x<1$ we have $f^{n}(x) \rightarrow 0$ as $n \rightarrow \infty$.

(ii) Must it be the case that for every $\epsilon>0$ there exists $n$ with the property that $f^{n}(x)<\epsilon$ for all $0<x<1$ ? Justify your answer.

Now suppose that we remove the condition that $f$ be continuous.

(iii) Give an example to show that it need not be the case that for every $0<x<1$ we have $f^{n}(x) \rightarrow 0$ as $n \rightarrow \infty$.

(iv) Must it be the case that for some $0<x<1$ we have $f^{n}(x) \rightarrow 0$ as $n \rightarrow \infty$ ? Justify your answer.

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