Problem:

Suppose that $\displaystyle { u_{k} }$ is a sequence of points in $\displaystyle \mathbb{R}^n $ that converges to the point u and that $\displaystyle \| u \| = r > 0 $. Prove that there is an indexKsuch that $\displaystyle \| u_{k} \| > \frac{r}{2} $ if $\displaystyle k \geq K $

===========================

Attempt:

By given, $\displaystyle \lim_{k \to infty} u_{k} = u $ and $\displaystyle \| u \| = r >0 $.

I believe I have to use following criterion, but do not know how to apply it to prove the problem.

Definition:A sequence of points $\displaystyle { u_{k} }$ in $\displaystyle \mathbb{R}^n $ is said to converge componentwise to the point u for each indexiwith $\displaystyle 1 \leq i \leq n $, then

$\displaystyle \lim_{k \to infty} p_{i}(u_{k}) = p_{i}(u) $.

Componentwise Convergence Criterion:Let $\displaystyle { u_{k} }$ be a sequence in $\displaystyle \mathbb{R}^n $ and let $\displaystyle u \in \mathbb{R}^n $. Then $\displaystyle { u_{k} }$ converges to u if and only if $\displaystyle { u_{k} }$ converges componentwise to u.

Thank you for your help.