Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space. Let $(A_n)_{n\geq1}$ be a sequence of events in $\mathcal{F}$. Prove that $\mathbb {P}(\limsup_{n \geq 1} A_n) = 0$ if $\sum_{i=1}^\infty \mathbb{P}(A_i)$ converges.
My attempt:
$$0 \leq\mathbb {P}(\limsup_{n \geq 1} A_n) = \lim_{n \to \infty}\mathbb {P}(\bigcup_{k=n}^\infty A_k)\leq \lim_{n \to \infty}\sum_{k=n}^\infty \mathbb{P}(A_k) = 0$$
The last equality needs some work. The other (in)equalities are basic properties.
Let $\sum_{i=1}^\infty \mathbb{P}(A_i):= S$. Let $\epsilon > 0$. Choose $n_0$ such that $n \geq n_0$ implies that $|S - \sum_{i=1}^n \mathbb{P}(A_i)| < \epsilon$. Then, if $n \geq n_0+1$, it follows that $|\sum_{k=n}^\infty \mathbb{P}({A_k})| = |\sum_{k=1}^ \infty\mathbb{P}(A_k) - \sum_{k=1}^{n-1}\mathbb{P}(A_k)| = |S - \sum_{k=1}^{n-1}\mathbb{P}(A_k)| < \epsilon$. This proves the desired equality. Is this correct?
The last equality is an elementary result from calculus: Set $a: = \sum_{i=1}^\infty \Bbb P (A_i)$. Then $$ \lim_{n \to \infty} \sum_{i=n}^\infty \Bbb P (A_i) = \lim_{n \to \infty} \sum_{i=1}^\infty \Bbb P (A_i) - \sum_{i=1}^{n-1} \Bbb P (A_i) =a -\lim_{n \to \infty} \sum_{i=1}^{n-1} \Bbb P (A_i) = a-a =0. $$