Property of sequences in $\mathbb{R}^N$

Question

Can someone help me with the proof of this : " Let $(x_n)\subset \mathbb{R}^N$ , then for a subsequence still denoted by $(x_n)$, $$ \begin{cases} |x_n|\to+\infty \\ \text{or}\\ |x_n|\to x\in\mathbb{R} \end{cases} $$

Thank you

Answer 1

If no subsequence of $(|x_n|)_{n\in\mathbb N}$ goes to $+\infty$, then the sequence $(|x_n|)_{n\in\mathbb N}$ is bounded and therefore, by the Bolzano-Weiertrass theorem, it has a convergent subsequence.

Answer 2

If $x_n$ is bounded then it follows by Bolzano-Weierstrass there is converging subsequence $(x_{n_k})_k$ to converging to some $a$ hence $$|x_{n_k}|\to |a|$$

if not $x_n$ is unbounded that is for each $k$ there $n_k$ such that $$|x_{n_k}|\ge k$$ hence$$\lim_{k\to\infty}|x_{n_k}|\ge \lim_{k\to\infty}k=\infty$$