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I am solving Ex.3.4.10 in Durrett's book. Given $\{X_n\}$ is a sequence of independent random variables with $|X_n|\leq M$ and $\sum_n var(X_n) = \infty$. Define $S_n = X_1 + X_2 +.... +X_n$, then prove that

$$(S_n - E S_n)/ \sqrt{var(S_n)}$$ converges to $N(0,1)$ in distribution.

My try: I am following the proof of the central limit theorem to prove this problem. However, I don't know how to use the assumption $|X_n|\leq M$. Could anyone please help me interpret this assumption? Thank you very much?

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    Hint: You could use the Lindeberg-Feller theorem(Th. 3.4.5). – 2017-02-28

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