### Question

Let $X$ be some reflexive Banach space. Suppose $x_n$ is some sequence in $X$ that weak converges to some $y \neq 0$. Is it the case that $$ \limsup \|x_n - y\| < \limsup \|x_n\| ?$$

### Positive with Hilbert spaces

In the Hilbert space case this is true, as $$ \langle x_n - y, x_n -y \rangle = \| x_n\|^2 + \|y\|^2 - \langle x_n, y\rangle - \langle y, x_n\rangle $$ and by weak convergence the latter two terms both converges to $\|y\|^2$ and we get $$ \limsup \|x_n - y\|^2 = \limsup \|x_n\|^2 - \|y\|^2. $$

### A negative example with a non-reflexive Banach space and weak* convergence

Let $X = L^\infty(\mathbb{R})$, and take $x_n = \mathbf{1}_{[-2,-1]} + \mathbf{1}_{[n,n+1]}$. Then $x_n$ weak* converges to $\mathbf{1}_{[-2,-1]}$ but $\|x_n - \mathbf{1}_{[-2,-1]}\|_{\infty} = 1 = \|x_n\|_\infty$.