Show that in the space of all sequences $s$ with the metric $\sum_{j\geq1}\frac{1}{2^j} \frac{|\xi_j^n-\xi_j|}{1+|\xi_j^n-\xi_j|}$ we have $x_n \rightarrow x $ if and only if $\xi_j^n \rightarrow \xi_j$ for all j = 1, 2, ... , where $x_n = (\xi_j^n) $ and $x = (\xi_j)$.
How to prove the reverse: that pointwise convergence implies that the sequence of series converges. (Kreyzsig book says it's trivial, but I don't think I can't exchange the iterated limits without some assumptions). Thanks.
Let $\epsilon >0$. Choose $N$ such that $\sum\limits_{k=N+1}^{\infty} \frac 1 {2^{k}} <\epsilon$. [Possible because $\sum\limits_{k=1}^{\infty} \frac 1 {2^{k}} <\infty$]. Now choose $n_0$ such that the first $N$ terms of the series are less than $\epsilon$ for $n \geq n_0$. [ Possible because of the terms has limit $0$]. Now conclude.