Fake pf: Let $X$ a Banach space, then let $x_n$ converge weakly to $x$, then as $X^* $ norms $X$ it follows that there exits some $ f \in X^*$ such that
$$ ||x_n - x|| = f(x_n-x) = f(x_n) - f(x) $$
by weak convergence $f(x_n) \to f(x) \implies f(x_n) - f(x) \to 0 $
I know this is wrong since the standard schauder basis in $\ell^p$ converges weakly but not strongly, I just do not know where the proof above went wrong.
Thank you!