| course |
course_year |
question_number |
tags |
title |
year |
Analysis II |
IB |
17 |
|
$3 . \mathrm{II} . 11 \mathrm{E} \quad$ |
2002 |
Show that if $a, b$ and $c$ are non-negative numbers, and $a \leqslant b+c$, then
$$\frac{a}{1+a} \leqslant \frac{b}{1+b}+\frac{c}{1+c}$$
Deduce that if $(X, d)$ is a metric space, then $d(x, y) /[1+d(x, y)]$ is a metric on $X$.
Let $D={z \in \mathbb{C}:|z|<1}$ and $K_{n}={z \in D:|z| \leqslant(n-1) / n}$. Let $\mathcal{F}$ be the class of continuous complex-valued functions on $D$ and, for $f$ and $g$ in $\mathcal{F}$, define
$$\sigma(f, g)=\sum_{n=2}^{\infty} \frac{1}{2^{n}} \frac{|f-g|{n}}{1+|f-g|{n}}$$
where $|f-g|{n}=\sup \left{|f(z)-g(z)|: z \in K{n}\right}$. Show that the series for $\sigma(f, g)$ converges, and that $\sigma$ is a metric on $\mathcal{F}$.
For $|z|<1$, let $s_{k}(z)=1+z+z^{2}+\cdots+z^{k}$ and $s(z)=1+z+z^{2}+\cdots$. Show that for $n \geqslant 2,\left|s_{k}-s\right|_{n}=n\left(1-\frac{1}{n}\right)^{k+1}$. By considering the sums for $2 \leqslant n \leqslant N$ and $n>N$ separately, show that for each $N$,
$$\sigma\left(s_{k}, s\right) \leqslant \sum_{n=2}^{N}\left|s_{k}-s\right|_{n}+2^{-N},$$
and deduce that $\sigma\left(s_{k}, s\right) \rightarrow 0$ as $k \rightarrow \infty$.