Show that a set is Borel Let $\mathbb R^\infty$ be the set of all sequence of real numbers endwed with the product topology associated to the distance $d(x,y)= \sum_{i=1}^\infty 2^{-n} \lvert x_n - y_n \rvert (1+ \lvert x_n -y_n \rvert )^{-1}$. 
Let $l^2$ be the space of all the sequence $\{x_h\}_h $ of real numbers such that $\sum_{h=1}^\infty x_h^2 < \infty $, endowed with the scalar product $(x,y)=\sum_{i=1}^\infty x_i y_i $ with $x,y \in l^2$. 
I have to show that $l^2$ is a Borel set in $\mathbb R^\infty$. 
Any hint? 

Hints: (1) For each $n$
the map $s_n:x\mapsto \sum_{h=0}^n x_h^2$, of $\Bbb R^\infty$ to $[0,\infty)$, is continuous, hence Borel measurable; (2) $s(x):=\uparrow\lim_n s_n(x)$ is also Borel, as a mapping of $\Bbb R^\infty$ into $[0,\infty]$; (3) express $l^2$ as the inverse image of a Borel set under a Borel map. 
