Let $(E_n)$ be a sequence of events in a probability space such that $$\lim_{n\to\infty}\mathbb P(E_n)=0.$$ I am trying to prove that if $$\sum_{n=1}^\infty\mathbb P (E_n\setminus E_{n+1}) <\infty$$ then $$\mathbb P\left(\limsup_{n\to\infty} E_n\right)=0.$$
For an example I am using the sequence of intervals $[0, 1/n]$ which goes to $0$ as $n$ increases to infinity. Then I can see how the probability of infinite sum of the intersection of the $E_n$ and the complement of $E_{n+1}$ is less than infinity and the probability of $\limsup_{n\to\infty} E_n$ equals zero.
Can anyone help me figure out how to prove this in general using any sequence? Especially to show how $\mathbb P\left(\limsup_{n\to\infty} E_n\right)=0$?
Thank you in advance for your thoughts.
Note that $$ \bigcup_{n=m}^{\infty} E_n = \bigcup_{n=m}^{\infty} E_n\setminus E_{n+1} $$ Let $\epsilon > 0$, since $\sum P(E_n\setminus E_{n+1}) < \epsilon$, then $\exists m\in \mathbb{N}$ such that $$ \sum_{n=m}^{\infty} P(E_n\setminus E_{n+1}) < \epsilon $$ and so $$ P(\bigcup_{n=m}^{\infty} E_n) < \epsilon $$ So if $E = \limsup E_n$, then $P(E) <\epsilon$. This is true for any $\epsilon >0$, so $P(E) = 0$.