The question is:
Consider $n$ Bernoulli trials, where for $i = 1, 2,..., n$, the $i$th trial has probability $p_i$ of success, and let $X$ be the random variable denoting the total number of successes. Let $ p \ge p_i$ for all $i = 1, 2, \ldots , n$. Prove that for $ 1 \le k \le n$,
$\Pr \{ X < k \} \ge \sum_{i=0}^{k-1}b(i; n, p)$
I tried to use induction on $k$ but obviously it doesn't work.