The Bonferroni correction is a single-step procedure that controls the family-wise error rate when testing $m$ null hypotheses simultaneously. Each individual hypothesis $H_i$ is tested at the adjusted significance level $\alpha / m$, equivalently by comparing its raw $p$-value $p_i$ to $\alpha / m$. By Boole's (Bonferroni) inequality, $\Pr\!\left(\bigcup_{i=1}^{m} \{p_i \leq \alpha/m\}\right) \leq \sum_{i=1}^{m} \Pr(p_i \leq \alpha/m) \leq \alpha$ under any joint distribution of the test statistics, so $\text{FWER} \leq \alpha$ without dependence assumptions. The correction is conservative because the bound is loose when tests are positively dependent or when many nulls are false, which is the gap that step-down refinements such as Holm-Bonferroni close.