Paper 1, Section II, E

Analysis I | Part IA, 2012

(a) What does it mean to say that the sequence (xn)\left(x_{n}\right) of real numbers converges to R?\ell \in \mathbb{R} ?

Suppose that (yn(1)),(yn(2)),,(yn(k))\left(y_{n}^{(1)}\right),\left(y_{n}^{(2)}\right), \ldots,\left(y_{n}^{(k)}\right) are sequences of real numbers converging to the same limit \ell. Let (xn)\left(x_{n}\right) be a sequence such that for every nn,

xn{yn(1),yn(2),,yn(k)}x_{n} \in\left\{y_{n}^{(1)}, y_{n}^{(2)}, \ldots, y_{n}^{(k)}\right\}

Show that (xn)\left(x_{n}\right) also converges to \ell.

Find a collection of sequences (yn(j)),j=1,2,\left(y_{n}^{(j)}\right), j=1,2, \ldots such that for every j,(yn(j))j,\left(y_{n}^{(j)}\right) \rightarrow \ell but the sequence (xn)\left(x_{n}\right) defined by xn=yn(n)x_{n}=y_{n}^{(n)} diverges.

(b) Let a,ba, b be real numbers with 0<a<b0<a<b. Sequences (an),(bn)\left(a_{n}\right),\left(b_{n}\right) are defined by a1=a,b1=ba_{1}=a, b_{1}=b and

 for all n1,an+1=anbn,bn+1=an+bn2\text { for all } n \geqslant 1, \quad a_{n+1}=\sqrt{a_{n} b_{n}}, \quad b_{n+1}=\frac{a_{n}+b_{n}}{2} \text {. }

Show that (an)\left(a_{n}\right) and (bn)\left(b_{n}\right) converge to the same limit.

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