probability - Proof explanation - weak law of large numbers - Mathematics Stack Exchange

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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Law of Large Numbers Strong and weak, with proofs and exercises

Law of Large Numbers Strong and weak, with proofs and exercises