Let $(X\_n)\_{n\ge 1}$ be a sequence of mutually independent random variables, on the same probability space, with expectation 0 and finite variance. Let $S\_n = \sum\_{l=1}^n X\_l$. Prove that for any $\lambda > 0$,

$$\lambda^2 \displaystyle P \left(\max\_{1\le k\le n} \big| S\_k \big| \ge \lambda \right) \le Var(S\_n)$$

Prove that if $\sum\_l Var(X\_l) < \infty$, then $(S\_n)\_{n\ge 1}$ converges almost surely. 

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