Convergent sequence on unit sphere Suppose $x_n$ is a bounded sequence in a vector space $V$ with norm $||\cdot||$. Show that if:
$$\hat{x}_n=\frac{x_n}{||x_n||}\;\;\text{converges}\Rightarrow x_n\;\text{has a convergent subsequence}$$

Thoughts: Writing the limit on the sphere as $\hat{x}=x/||x||$ I would like to show that for large enough $M$ (bounding $||x_n||$), there are $n_0<n_1<\cdots$ such that
$$\frac{1}{M}||x_{n_k}-x||\leq ||\hat{x}_{n_k}-\hat{x}||\quad \forall k>K$$
First, is this even true? If so could can I show it? 

Hint: The sequence of numbers $\| x_n\|$ is real and bounded, so it must contain a convergent subsequence by the Bolzano–Weierstrass theorem. 
