Convergence of vectors in a normed vector space.

Question

I am seeking advice and comments on the following proof and how to improve it, as I have struggled with proofs involving limits.

Given: Let ($V$,|| ||) be an abstract normed vector. Suppose $V$ contains a sequence of vectors, $\{a_n:n\in \mathbb{N}\}$, which converges to some vector $u$ in $V$.

Claim: $\lim_{n \rightarrow \infty}(||a_n||-||u||)=0$

Proof: $\forall \epsilon >0, \exists$ an integer $N>0$ such that $||a_n-u||<\epsilon$ whenever $n>N$. Then when $n>N, ||a_n-u||<\epsilon$, implying $||a_n||=||u||$. Thus $\{||a_n||-||u||\}=0$.

Any comments are appreciated.

Answer 1

Hint You just need to show that $$\left|\|x\|-\|y\|\right|\leq\|x-y\|$$ for all $x,y\in V$.