Let $\{T_n\}$ random variables independent and equally distributed, at $(\Omega,\mathcal{F}, P)$, with $0 < m :=E(T_1) < 1$.
(a) Let $0 < \epsilon < m$. Prove thath for almost every $\omega \in \Omega$ exists $N_{\omega} \in \mathbb{N}$ such that $$n(m - \epsilon) < (T_1 + T_2 + \ldots + T_n)(w) < n(m + \epsilon) \ \ \ \ \ \ \ \ \forall n \geq N_{\omega}.$$
(b) Prove that $$P\left[\sum_{n=1}^{\infty}T_n < \infty\right]=0.$$
I have problems with the second exercise. The first exercise I know it is consequence of the strong law of large number since $E[X_1]<\infty$, and because we have $$ |\overline{T}(\omega)-E[T_1]| < \epsilon.$$
But the second exercise I have no idea what is going on... In fact I do not know if I understand the exercise.
Any hints?
By the strong law of large numbers, $\frac1k\sum_{n=1}^k T_n\to m$ almost surely as $k\to\infty$, i.e.
$$\mathbb P\left[\frac1k \sum_{n=1}^k T_n \overset{k\to\infty}\longrightarrow m\right]=1$$ which implies part (a) and (b) which is equivalent to $$\mathbb P\left[\sum_{n=1}^\infty T_n =+\infty\right]=1$$ Since for the sample average to converge, the sum must go to infinity (since the denominator goes to infinity)