Chazelle's discrepancy book: greedy method
My oversight was that $|S| = m$, so there are $m$ events of the form "$S_i$ is bad".  Then since $Pr[\cup_{i=1}^m A_i] \le \sum_{i=1}^m Pr[A_i]$ for any collection of events $\{A_i \mid i=1,2,\dots,m\}$, it follows that $Pr[\text{some }S_i\text{ is bad}] &lt; 1$, so $Pr[\text{no }S_i\text{ is bad}] &gt; 0$.