A non weakly convergent sequencein $\ell_2$ I'm going to prove that this sequence $(x_n=ne_n)$ where $e_n$ is $0$ in all place and $1$ in n'th, is a non-weakly convergent sequence in $\ell_2$, I know that if $x_n$ weakly convergence then it will be norm bounded, it' clear that itis not, so this sequence does not converge weakly, but I want to prove this fact by considering the function $x^*=(1,\frac{1}{2},\frac{1}{3}\ldots)$ in $\ell_2$. Any help is appreciated.

It is not possible to disprove the weak convergence $x_n \rightharpoonup x$ (for some $x \in \ell_2)$ by _only_ looking at $x^*$.

Indeed, $(x_n , x^*) = 1$. Thus, you cannot disprove, e.g., $x_n \rightharpoonup x_1$ in $\ell_2$ by _only_ looking at $x^*$.

It is sufficient to additionally consider the unit sequences $e_n$. Let us assume that $x_n \rightharpoonup x$ for some $x \in \ell_2$. From $(x_n, e_n) \to 0$, you learn something about $x$ and this contradicts $(x_n, x^*) \to (x,x^*)$.