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

The weak law of large numbers is a very strong

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Solved (15pt) Recall the weak law of large numbers of the

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

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