probability - Proof explanation - weak law of large numbers

Let $(X_i)$ be i.i.d. random variables with mean $\mu$ and finite variance. Then $$\dfrac{X_1 + \dots + X_n}{n} \to \mu \text{ weakly }$$ I have the proof here: What I don't understand is, why it

i.ytimg.com/vi/Yh5bR7X3ch8/sddefault.jpg

Weak Law of Large Numbers (WLLN). Overview, by Pablo Kowalski Kutz

Law of Large Numbers - Definition, Example, How to Use

Law of Large Numbers: What It Is, How It's Used, Examples

Law of Large Numbers: What It Is, How It's Used, Examples

PDF) On a survey of uniform integrability of sequences of random variables.

i.ytimg.com/vi/uQTFiXQR4PQ/hq720.jpg?sqp=-oaymwEhC

12 Probability and applications

Law of large numbers - Wikipedia

Law of large numbers - Wikipedia

Law of Large Numbers - Definition, Examples, Insurance, Statistics

Law of large numbers - Wikipedia

A Gentle Introduction to the Law of Large Numbers in Machine Learning

What Is the Law of Large Numbers? (Definition)