Consider $n$ independent events $A_1$, $A_2$, $\ldots$, $A_n$, let $P(A_k)=p_k$ and $p = p_1+p_2+\cdots p_n$, then the probability that at least $k$ of the events $A_1$, $A_2$, $\ldots$, $A_n$ happen is less than $p^k/k!$.
I have noticed that:
$$ \frac{p^k}{k!}>\sum p_{i_1}p_{i_2}\cdots p_{i_k} $$ where $i_k\in\{1,2,\cdots n\}$ and $i_1<i_2<i_3\cdots<i_k$.
However I don't know how to proceed.