3.II.11E3 . \mathrm{II} . 11 \mathrm{E} \quad

Analysis II | Part IB, 2002

Show that if a,ba, b and cc are non-negative numbers, and ab+ca \leqslant b+c, then

a1+ab1+b+c1+c\frac{a}{1+a} \leqslant \frac{b}{1+b}+\frac{c}{1+c}

Deduce that if (X,d)(X, d) is a metric space, then d(x,y)/[1+d(x,y)]d(x, y) /[1+d(x, y)] is a metric on XX.

Let D={zC:z<1}D=\{z \in \mathbb{C}:|z|<1\} and Kn={zD:z(n1)/n}K_{n}=\{z \in D:|z| \leqslant(n-1) / n\}. Let F\mathcal{F} be the class of continuous complex-valued functions on DD and, for ff and gg in F\mathcal{F}, define

σ(f,g)=n=212nfgn1+fgn\sigma(f, g)=\sum_{n=2}^{\infty} \frac{1}{2^{n}} \frac{\|f-g\|_{n}}{1+\|f-g\|_{n}}

where fgn=sup{f(z)g(z):zKn}\|f-g\|_{n}=\sup \left\{|f(z)-g(z)|: z \in K_{n}\right\}. Show that the series for σ(f,g)\sigma(f, g) converges, and that σ\sigma is a metric on F\mathcal{F}.

For z<1|z|<1, let sk(z)=1+z+z2++zks_{k}(z)=1+z+z^{2}+\cdots+z^{k} and s(z)=1+z+z2+s(z)=1+z+z^{2}+\cdots. Show that for n2,sksn=n(11n)k+1n \geqslant 2,\left\|s_{k}-s\right\|_{n}=n\left(1-\frac{1}{n}\right)^{k+1}. By considering the sums for 2nN2 \leqslant n \leqslant N and n>Nn>N separately, show that for each NN,

σ(sk,s)n=2Nsksn+2N,\sigma\left(s_{k}, s\right) \leqslant \sum_{n=2}^{N}\left\|s_{k}-s\right\|_{n}+2^{-N},

and deduce that σ(sk,s)0\sigma\left(s_{k}, s\right) \rightarrow 0 as kk \rightarrow \infty.

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