Let $X$ be a separable space with infinite dimension, let $(\cdot,\cdot)$ and $\|\cdot \|$ be the scalar product and the norm of $X$, and $\{e_n\}_n$ be an orthonormal basis of $X$. We define $\|x\|_0^2=\sum_{n=1}^\infty\frac{|(x,e_n)|^2}{n^2}.$
What I already proved is that $\|\cdot \|_0$ is a norm and that the set $\{x\in X : \|x\|\leq 1\}$ is compact in $(X,\|\cdot\|_0)$; I am then asked to prove that $(X,\|\cdot\|_0)$ is not complete. So I focused on trying to find a Cauchy sequence in $(X,\|\cdot\|_0)$ not converging to an element in this space. By the previous point such a sequence $\{x_n\}_n$ should not be bounded with respect to the norm $\|\cdot\|$, but up 'till now I was not able to find such an example.
Can anybody help me?
Regards
-Guido-