A question on limitsup

Question

What is the relationship between $\lim \sup _{n}\left(a_{n} b_{n}\right)$ and $(\lim \sup _{n} a_{n})(\lim \sup _{n} b_{n})$?

Answer 1

If $a_n$, $b_n$ are negative, just consider instead $-a_n$, $-b_n$, so that they are $\geq 0$. As you can find, for instance, on wikipedia (https://en.wikipedia.org/wiki/Limit_superior_and_limit_inferior), it can be shown that: $$ \limsup_{n \rightarrow + \infty} (a_n b_n) \leq (\limsup_{n \rightarrow + \infty} a_n) \cdot (\limsup_{n \rightarrow + \infty} b_n) $$ while $$ \liminf_{n \rightarrow + \infty} (a_n b_n) \geq (\liminf_{n \rightarrow + \infty} a_n) \cdot (\liminf_{n \rightarrow + \infty} b_n)$$ (where we are assuming that no indeterminate forms arise from the products of the limits). You can prove these inequalities using some simple properties of $\sup$ and $\inf$ and the definitions of $\limsup$ and $\liminf$.