I know that a sequence is a convergent sequence if and only if it is a Cauchy sequence so lim inf $a_n$ = $\lim sup a_n$.
Then, if $\liminf a_n = \infty$, then can we conclude $\lim a_n=\infty$ ?
What about, if $\limsup a_n= -\infty$, then what can we conclude?
My understanding so far is, if $\liminf a_n=\infty$, then $\limsup a_n\neq \infty$, so $\lim a_n \neq \infty$.
I am not sure about the second if statement.
Best Answer
If $\liminf_na_n=\infty$, then, since$$\liminf_na_n=\lim_n\inf_{k\geqslant n}a_k,$$you have that, for any $M>0$, $\inf_{k\geqslant n}a_k>M$ if $n$ is large enough; in particular, $a_n>M$ if $n$ is large enough. In other words, $\lim_na_n=\infty$.
A similar argument shows that if $\limsup_na_n=-\infty$, then $\lim_na_n=-\infty$.