Suppose we run the algorithm for $n$ times and, therefore, the probability for all of these independent trials to fail is $(1-p)^{n}$.
The probability to success is, therefore:
$$
\begin{aligned}
1-(1-p)^{n} & \geq 1-\epsilon \\
(1-p)^{n} & \leq \epsilon \text { (flipping inequality due to change in sign) }
\end{aligned}
$$
