Given the sequence $a_n = n(2+(-1)^n)$.
Find $\mathop{\overline{\lim}}\limits_{n\to\infty} a_n$ and $\mathop{\underline{\lim}}\limits_{n\to\infty} a_n$.
The following is how I approach this problem.
$\mathop{\lim}\limits_{n\to\infty} a_{2n} = \mathop{\lim}\limits_{n\to\infty} n(2+1) = \infty$
and
$\mathop{\lim}\limits_{n\to\infty} a_{2n+1} = \mathop{\lim}\limits_{n\to\infty} n(2-1) = \infty$
Could I conclude $\mathop{\overline{\lim}}\limits_{n\to\infty} a_n = \mathop{\underline{\lim}}\limits_{n\to\infty} a_n = \infty$?
I'm confused because $\infty$ is not a constant.
What's the problem with the fact that $\infty$ is not a constant? For each $n\in\Bbb N$, $a_n\geqslant n$, and therefore $\lim_{n\to\infty}a_n=\infty$. So, $\limsup_{n\to\infty}a_n=\liminf_{n\to\infty}a_n=\infty$.
In general, if $l\in\Bbb R\cup\{\pm\infty\}$, then$$\lim_{n\to\infty}a_n=l\iff\limsup_{n\to\infty}a_n=\liminf_{n\to\infty}a_n=l.$$