I provide simplified proofs for each of the following fundamental theorems regarding selection principles:

1. (1) The Quasinormal Convergence Theorem, due to the author and Zdomskyy, asserting that a certain, important property of the space of continuous functions on a space is actually preserved by Borel images of that space.

2. (2) The Scheepers Diagram Last Theorem, due to Peng, completing all provable implications in the diagram.

3. (3) The Menger Game Theorem, due to Telgársky, determining when Bob has a winning strategy in the game version of Menger’s covering property.

4. (4) A lower bound on the additivity of Rothberger’s covering property, due to Carlson.

The simplified proofs lead to several new results.

We establish a surprising connection between Menger's classical covering property and Blass-Laflamme's modern combinatorial notion of groupwise density. This connection implies a short proof of the groupwise density bound on the additivity number for Menger's property.

We answer a question of Just, Miller, Scheepers and Szeptycki whether certain diagonalization properties for sequences of open covers are provably closed under taking finite or countable unions.