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> Let $(X\_n)$ a sequence of i.i.d random variables, such that $P(X\_1 \geq 0)=1$ and $P(X\_1 >0) > 0$. Show that $\sum\_{n \geq 1}X\_n =\infty$ almost surely.
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We have that $P(X\_1 > 0)= \underset{n \rightarrow \infty}{\lim}P(X\_1 > \frac{1}{n})$. So by the property of the limit, there exists a sufficiently big $N \in \mathbb{N}$ such that $\forall n \geq N$ we have $P(X\_1 > \frac{1}{n})>0$. Let then put $A\_n = \{X\_1 \geq \frac{1}{N} \}$. We then have:

$$\sum\_{n \geq 1} X\_n \geq \sum P(A\_n) \geq M + \sum\_{n\geq N} P(X> \frac{1}{N}) = M + P(X>\frac{1}{N})\sum\_{n\geq N}1$$
Which equals to infinity. Is my reasoning correct?

I am very unsure, as $\sum\_{n\geq 1}X\_n$ is ambiguous to me, I am unsure what it exactly represents.