William Trench , Introduction to Real Analysis Page 187.
Limit superior and inferior:
When he said in (a) (if the sequence is bounded above and does not diverge to minus infinity). Does not it mean that the sequence is bounded above and below?
The same for the (b)
Not diverging to $-\infty$ does not guarantee that the sequence is bounded below.
E.g. take the sequence $(s_n)_n$ taking value $0$ for odd $n$ and value $-n$ for even $n$.
We do not have $\lim_{n\to\infty} s_n=-\infty$ (no divergence to $-\infty$) but the sequence is not bounded below either.