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

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Proof of Strong law of large numbers and weak law of large numbers

Stats Syllabus, PDF, Probability Theory

Proof of the Law of Large Numbers Part 1: The Weak Law, by Andrew Rothman

Unraveling the Law of Large Numbers, by Sachin Date

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

real analysis - Proof of the strong law of large numbers for bernoulli random variables - Mathematics Stack Exchange

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Proof of the Law of Large Numbers Part 2: The Strong Law, by Andrew Rothman