When limit superior diverges

Quote:

Originally Posted by Connected

Prove that $\lim\sup x_n=\infty\iff \forall M>0,\forall n\in\mathbb N,\exists k_0\ge n$ so that $x_k>M.$

I think is an easy problem, but I'm confused, the statement establishes that $x_k$ is bounded below, but not above.

How to prove this?

Which part are you having trouble with? Try firstly the only if statement. Then, you want to prove that $\limsup x_n=\infty$ but evidently by assumption you have that $\displaystyle \sup_{n\geqslant N}x_n\geqslant M$ for every $M\in\mathbb{R}^+$ so that $\displaystyle \limsup x_n=\lim_{N\to\infty}\sup_{n\geqslant N}x_n\geqslant M$ from where the conclusion follows since $M$ were arbitrary (intuitively you can take the limit as $M\to\infty$ of both sides of this last expression, but the left side is 'unaffected' by the limit since there is no $M$ )