Product of lim sups

Question

Suppose lim sup $a_n$ is finite, and $c_n \to c$

Prove that if $c \geq 0$ lim sup $a_n c_n$ = c lim sum $a_n$ and find a counterexample to this if $c <0$.

Is there a rule that the product of lim sups is equal to the lim sup of the product? Also, what counterexample will work here?

Answer 1

Let $c_n=-1$ for all $n$. Try to prove this:

$$\limsup_{n\to\infty}(-a_n)= = -\liminf_{n\to\infty}a_n $$

For general proof of positive $c$:

Since $c_n \rightarrow c$, then $\forall \alpha>0$, $\exists N>0$ such that $0<c-\alpha<c_n<c+\alpha$. Then $$(c-\alpha)\sup_{n>N} a_n <\sup_{n>N} c_n a_n< (c+\alpha)\sup_{n>N} a_n$$