Sum of infinitely many i.i.d. random variables is infinite with probability 1

Question

How do I solve this? I'm really confused.

If $X_1,X_2,\ldots$ are non-negative independently and identically distributed random variables with $P(X_i>0)>0$, show that $\displaystyle P\left(\sum_{i=1}^\infty X_i=\infty\right)=1$.

Answer 1

Hint: It suffices to show for each $n$ that $$P\bigg\{\omega:\sum_{i=1}^\infty X_i(\omega)>n\bigg\}=1,$$ which is the case iff for all $0<\delta<1$ $$P\bigg\{\omega:\sum_{i=1}^\infty X_i(\omega)>n\bigg\}>\delta.$$ There is $\epsilon>0$ such that $P(X_i>\epsilon)>0$.

Now for each $m$ we have $$P\bigg\{\omega:\sum_{i=1}^m X_i(\omega)>n\bigg\}\leq P\bigg\{\omega:\sum_{i=1}^\infty X_i(\omega)>n\bigg\},$$ so it suffices to show, using the $\epsilon$, that for some $m$ large enough, $$P\bigg\{\omega:\sum_{i=1}^m X_i(\omega)>n\bigg\}>\delta,$$ which is an elementary argument. Try it.