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.

我为以下每个有关选择原则的基本定理提供了简化的证明：

1. (1)拟正规收敛定理，由作者和 Zdomskyy 提出，断言空间上连续函数的空间的某个重要性质实际上由该空间的 Borel 图像保留。

2. (2) Scheepers 图最后定理，由 Peng 完成，完成了图中所有可证明的含义。

3. (3)门格尔博弈定理，由 Telgársky 提出，确定鲍勃在门格尔覆盖属性的游戏版本中何时具有获胜策略。

4. (4)由 Carlson 提出的 Rothberger 覆盖特性的可加性下界。

简化的证明导致了一些新的结果。