How to obtain the formula for probability of at least m of the n events, $(A_1, \ldots ,A_n)$ occurring?

Question

In our textbook (William Feller) the formula for at least $m$ of the $n$ events, $(A_1, \ldots ,A_n)$ occurring simultaneously is given by:$$P_m = S_m - \binom{m}{1}S_{m+1}+\binom{m+1}{2}S_{m+2}-\dots\pm\binom{n-1}{m-1}S_n$$ where $$S_k = \sum_{1\leq i_1< i_2<\dots< i_k\leq n}P(A_{i_1}\cap A_{i_2}\cap\dots \cap A_{i_k})$$ But how do I prove this? It looks like the inclusion-exclusion principle. But I cannot proceed in any useful way from the inclusion-exclusion principle.