$\liminf s_n = \limsup s_n = +\infty$ implies that the limit of the sequence is also $+\infty$

Question

I am trying to show that the fact that $\limsup (s_{n}) = \liminf (s_{n})=+\infty$ implies that $\lim(s_n)=+\infty$. Is the following correct?

$ \liminf(s_n)=+\infty$, so there exists some $B$, such that $n>B$ implies that $\inf\{s_n:n>B\}>M$, for all $M$. So that means that $n>B$ implies that $s_n>M$, and so we have shown that the sequence $s_n$ satisfies the definition of an infinite limit.

Thank you.

Answer 1

It is correct. In fact, it suffices to assume that $\liminf(s_n) = \infty$.(As José pointed out, you didn't use the $\limsup$ condition in the correct prove you gave.)