Counterexample for limsup

Question

Statement: $\limsup\limits_{n\to\infty} c_n a_n = c \limsup\limits_{n\to\infty} a_n$

Please help find a counterexample to this statement if $c<0$.

Edit: also suppose $c_n \to c$ and $\limsup a_n$ is finite

Answer 1

Let $c_n = -1 \to -1 = c $, and $a_n = (-1)^n$. then

$$ limsup (c_na_n) = 1$$

$$ -1 ( \limsup(a_n) ) = -1 $$

Answer 2

$c_n\equiv-1$, $a_n=\sin\left(\frac{n\pi}{6}\right)$.

Then $\limsup(c_na_n)=1$ but $\limsup a_n=1$ too hence $-\limsup a_n=-1\neq\limsup(c_na_n)=1$.