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

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

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SOLVED: Let X1, X2 be independent and identically distributed random variables with E(Xi) = p, and let Sn = Σ(Xi - p). Assume that the moment generating function Mx(t) of X exists

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