Prove the convergence of two sequences It is known that, $\sum_{i\in\mathbb{N}}\mathbb{P}[X_i \not= Y_i]< \infty$
As a consequence, I got to show the equivalence of
$\sum_{i\in \mathbb{N}} X_i(\omega)$ converges a.s. $\Leftrightarrow$ $\sum_{i\in \mathbb{N}} Y_i(\omega)$ converges a.s. 
where a.s. stands for almost surely.
Does the condition $\sum_{i\in\mathbb{N}}\mathbb{P}[X_i \not= Y_i]< \infty$ give me the convergence in probability (p) of $X_i$ and $Y_i$? In this case I would show that  if $X_i \xrightarrow{p} X$ and $Y_i \xrightarrow{p} Y$,then $X_i+Y_i \xrightarrow{p} X+Y$. Then I would prove the a.s. convergence with Ottaviani's inequality. Am I on the right track or is there a better way to prove?
 A: It's much easier: As
$$\sum_{i \geq 1} \mathbb{P}(X_i \neq Y_i)<\infty,$$
it follows from the Borel-Cantelli lemma that there exists for almost every $\omega \in \Omega$ a number $N \in \mathbb{N}$ such that
$$Y_i(\omega) = X_i(\omega) \qquad \text{for all $i \geq N$}.$$
In particular,
$$\sum_{i=N}^M X_i(\omega) = \sum_{i=N}^M Y_i(\omega)$$
for all $M \geq N$. Since the convergens of the series
$$\sum_{i \geq 1} X_i(\omega) \qquad \text{and} \qquad \sum_{i \geq 1} Y_i(\omega)$$
does not depend on the first $N$ terms, this proves that $\sum_{i=1}^{\infty} X_i(\omega)$ converges almost surely if and only if $\sum_{i=1}^{\infty} Y_i(\omega)$ converges almost surely.
A: Borel-Cantelli Lemma works just fine for this:
Borel-Cantelli Lemma: Let $(\Omega,\mathcal{M},\Bbb{P})$ be a probability space. Then 
$$\forall \{E_n\}_n\subseteq\mathcal{M}: \Sigma_n \Bbb{P}(E_n)<\infty \implies  \Bbb{P}\left(\limsup_n E_n\right)=0.$$

Recall that for a sequence $\{E_n\}_n$ of events,
$$\limsup_n E_n\stackrel{\tiny\mbox{def}}{=} \bigcap_n \bigcup_{k\geq n}E_k =\{\omega\in\Omega\vert \omega\in E_n \mbox{ for infinitely many } n\}$$

If we set $\forall n: E_n:=\{X_n\neq Y_n\}$, Borel-Cantelli Lemma tells you that for almost no outcome $\omega$ $\{X_n(\omega)\}_n$ and $\{Y_n(\omega)\}_n$ differ for infinitely many indices $n$.
Then the result follows by noticing that finitely many terms in a series do not affect convergence.
