Let $V$ be the space of real sequences {${x_{1},x_{2},...}$} so that $\sum_{k=1}^{\infty}x_{k}^{2}$ converges. Let $W\subset V$ be the set of rational sequences with a finite number of terms.
Let$<,>: V$x $V\to \mathbb R; <x,y>=\sum_{k=1}^{\infty}x_{k}y_{k}$(also converges)
$||x||=\sqrt{<x,x>}$ and a neighbourhood of radius $\epsilon\gt 0$ around $x\in V$ as {${y\in V : ||x-y||<\epsilon}$}
Prove that: $\forall \epsilon>0$, $\forall x\in V$ $\exists w\in W$ so that $||x-w||<\epsilon$
My attempt:
Let $\epsilon>0$ and $x\in V$; I know that $\sum_{k=1}^{\infty}x_{k}^{2}$ converges in other words: $\exists N>0$ (that depends on the $\epsilon$) so that $\forall m,n>N$,
$$|\sum_{k=1}^{n}x_{k}^{2}-\sum_{k=1}^{m}x_{k}^{2}|<\epsilon^{2}/2$$ if $m>n$ (withouth loss of generality) then
$$|\sum_{k=1}^{n}x_{k}^{2}-\sum_{k=1}^{m}x_{k}^{2}|=\sum_{k=n+1}^{m}x_{k}^{2}<\epsilon^{2}/2$$
We now that $\mathbb Q$ is dense in $\mathbb R$ in other words $\forall \epsilon>0$ $\forall x\in \mathbb R$, $\exists r\in \mathbb Q$ so that $|x-r|<\epsilon$
Hence we now that $$\exists w_{k}\in \mathbb Q$$ so that $$|x_{k}-w_{k}|<{\epsilon\over \sqrt{2(N+1)}}$$ $\forall k=1,2,...,N+1$
then $$\sum_{k=1}^{N+1}(x_{k}-w_{k})^{2}<\epsilon^{2}/2$$
On the other hand we now that $$\sum_{k=n+1}^{m}x_{k}^{2}<\epsilon^{2}/2$$ $\forall m>n>N$ then the minimum value that $n$ can take is $N+1$, and the minimum value that $n+1$ can take is $N+2$
$$\sum_{k=n+1}^{m}x_{k}^{2}=\sum_{k=N+2}^{m}x_{k}^{2}<\epsilon^{2}/2$$ taking the limit when $m\to \infty$
$$\lim_{m\to \infty}\sum_{k=N+2}^{m}x_{k}^{2}<lim_{m\to \infty}\epsilon^{2}/2$$ wich is equals to:
$$\sum_{k=N+2}^{\infty}x_{k}^{2}<\epsilon^{2}/2$$
Hence $$\sum_{k=1}^{N+1}(x_{k}-w_{k})^{2}+\sum_{k=N+2}^{\infty}x_{k}^{2}=\sum_{k=1}^{\infty}(x_{k}-w_{k})^{2}$$ (This is because we take $w_{j}=0 \forall j>N+1$
$$\sum_{k=1}^{\infty}(x_{k}-w_{k})^{2}<\epsilon^{2}/2+\epsilon^{2}/2=\epsilon^{2}$$
taking the square root: $$\sqrt{\sum_{k=1}^{\infty}(x_{k}-w_{k})^{2}}<\epsilon$$
but that is: $||x-w||<\epsilon$
I would like you to tell me if my proof is correct. Thank you :)