I'm to prove the following:
Let $H$ be a Hilbert space, and let $M$ be a non-empty convex subset of $H$.
Suppose that $(x_n)$ is a sequence in $M$ such that $ ||x_n|| \to d$, where $d= \inf \{||x||:x \in M \}$. Show that $(x_n)$ converges in $H$.
Is it enough to claim the following:
Let $d=||x||$. It follows from the continuity of $|| \cdot||$ that $$||x_n|| \to d \; \text{as} \; x_n \to x$$
Hence $(x_n)$ converges to some $x$ in $H$.
HINT:
Use the equality:
$$||\frac{x_n - x_m}{2}||^2 + ||\frac{x_n + x_m}{2}||^2= 1/2\,(\,||x_n||^2 + ||x_m||^2)$$
$M$ is convex so $\frac{x_n + x_m}{2}$ also have norm approaching $d$.