Convergence of the product of two Random Functions

Question

$lim_{t\to\infty}\mathbb{P}(|g(X(t))|>\epsilon)=0$,

$f(X(t))=\mathbb{1}_{({X(t)}\neq{0})}$

I want to show $f(X(t))g(X(t))$ coverage in probability to 0. Just wondering if it is possible, thank you.

Answer 1

$P(|f(X(t))g(X(t))| >\epsilon) \leq P(g(X(t))| >\epsilon)\to 0$. (I have just used the fact that $f$ is bounded by $1$).