Varied Type of Borel-Cantelli Lemma I

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Let $\{E_n\}$ be arbitrary events satisfying

(1) $\underset{n}{\lim}\mathscr{P}(E_n)=0$;
(2) $\underset{n}{\sum}\mathscr{P}(E_nE_{n+1}^c)<\infty$,

then $\mathscr{P}\{\limsup_n E_n\}=0$.

 $\bullet$ Proof.

For any $m$, we have
$$\begin{array}{rl}
\mathscr{P}\{\limsup_n E_n\}
 & =\mathscr{P}\left\{\bigcap_{m=1}^\infty\bigcup_{n\geq m}^\infty E_n\right\} \\
 & \leq\mathscr{P}\left\{\bigcup_{n\geq m}^\infty E_n\right\} \;(\because\mbox{ monotone}) \\
 & =\underset{N\rightarrow\infty}{\lim}\mathscr{P}\left\{\bigcup_{n\geq m}^N E_n\right\} \\
 & =\underset{N\rightarrow\infty}{\lim}\mathscr{P}\left\{\left(\bigcup_{n\geq m}^N E_nE_{n+1}^c\right)\cup E_N\right\} \;(\because\mbox{ see graphs below})\\
 & \leq\underset{N\rightarrow\infty}{\lim}\mathscr{P}\left\{\bigcup_{n\geq m}^N E_nE_{n+1}^c\right\}+\mathscr{P}\left\{E_N\right\} \;(\mbox{Boole's inequ.}) \\
 & \leq\underset{N\rightarrow\infty}{\lim}\sum_{n\geq m}^N \mathscr{P}\left\{E_nE_{n+1}^c\right\}+\mathscr{P}\left\{E_N\right\} \;(\mbox{Boole's inequ.}) \\
 & =\sum_{n\geq m}^\infty\mathscr{P}\left\{E_nE_{n+1}^c\right\} \rightarrow0\mbox{ as }m\rightarrow\infty.
\end{array}$$Since $\sum_n\mathscr{P}(E_nE_{n+1}^c)<\infty\implies\sum_{n\geq m}\mathscr{P}(E_nE_{n+1}^c)\rightarrow0$ as $m\rightarrow\infty$.

$\Box$