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 $k \rightarrow \infty$.