Let $l^2$ be the space of all real sequences $x = (x_1,x_2, x_3,\;...)$ for which $\sum_{n=1}^\infty |x_n|^2$ converges. It can be easily verified that map $$\langle x,y\rangle = \sum_{n=1}^\infty x_ny_n$$ defines an inner product on $l^2$ and that $x\to ||x||=\sqrt{\langle x,x\rangle}$ defines norm on $l^2$, therefore one can speak about Cauchy's sequences. I am aware of the fact that if the sequence $\{x^j\}_{j\in \mathbb{N}}$, $x^j\in l^2$ , converges, it must be Cauchy's. Clearly, the seqence $$x^1 =(1,0,0,0\;...)\\ x^2 = (0,1,0,0,\;...)\\x^3 = (0,0,1,0, \; ...)\\ \vdots$$ is not Cauchy's, since $||x^j-x^k||=\sqrt{2}$ for all $j\neq k$. As consequence, it cannot be convergent. On the other hand, we can notice that $$\forall n \in \mathbb{N} \forall j\in \mathbb{N}. n>j \Rightarrow x^j_n = 0\text{,}$$ so - intuitively - kind of limit exists and equals to $x = (0,0,0,\;...)$. My question is:
Is there any natural topology (or metric) for which the upper sequence really converges?
