Elementary proofs of four standard results on false discovery rate

We collect self-contained elementary proofs of four standard results in the literature on the false discovery rate of the Benjamini-Hochberg (BH) procedure for independent or positive-regression dependent p-values, the Benjamini-Yekutieli procedure for arbitrarily dependent p-values, and the e-BH procedure for arbitrarily dependent e-values. As a corollary, the above proofs also lead to some inequalities of Simes and Hommel.