Product of lim sups
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 $$(c-\alpha)\sup_{n>N} a_n <\sup_{n>N} c_n a_n< (c+\alpha)\sup_{n>N} a_n$$