Let $\epsilon > 0$ and $n, a \in \mathbb N$. Let $X$ be a random variable such that $$\mathbb P(X \leq \epsilon) \geq 1-a (1-\epsilon)^n.$$ I want to prove that $\mathbb E[X]$ is bounded by $\frac{\log(a)}{n}$ up to some constant factors.
This is my attempt
$$ \mathbb E[X] = \int^{\infty}_{0} \mathbb P(X > t)dt \leq \int^{\infty}_{0} a(1-t)^n dt.$$ Clearly the last integral diverges. I also thought of using $(1-t)^n \leq e^{-nt}$, but it also doesn't work. What should I do?