Answer
FALSE
Work Step by Step
Consider $a_{n}=(-1)^n$ and $b_{n}=(-1)^{n+1}$ (Both diverge)
Then
$a_{n}b_{n}=(-1)^n\times (-1)^{n+1}= (-1)^{2n+1}=-1$
Since only one of them is always negative for any $n$, it converges.
Hence, the statement is false.