So I have to prove the following excercise:
Let $N(t)=\sum_{n=1}^\infty \mathbb{1}_{\{J_n\leq t\}}$ where $J_n=\sum_{k=1}^n X_i$ where the $X_i$ are positive i.i.d. random variables.
Let $\{N(t),t>0\}$ be a renewal process, prove the followings equalities \begin{equation*} 1=\lim_{t\xrightarrow{}\infty}\mathbb{P}\left[N(t)>n\right]=\mathbb{P}\left[\lim_{t\xrightarrow{}\infty}N(t)>n\right], \end{equation*} so we can conclude that, $\mathbb{P}\left[\lim_{t\xrightarrow{}\infty}N(t)=\infty\right]=1$.
My problem is with the definition of Renewal Process. I claim that the above will be false if the $X_i$ aren't finite almost surely.