Borel-Cantelli lemma problem [duplicate]

Practice problem for exam:

Let ${A_n}$ satisfy $\sum\limits_{n=1}^\infty P(A_n \cap A^c_{n+1}) < \infty$ and $\lim\limits_{n\to \infty} P(A_n) = 0$. Show that $P(\lim \sup A_n) = 0$.

I can see that it is sufficient to show that $P\left(\sum\limits_{n=1}^\infty A_n\right) < \infty$ and then apply the Borel-Cantelli lemma, but I am having trouble showing this from the information given. I see an almost identical problem here: A variation of Borel Cantelli Lemma, however the hint there is not enough for me to make headway, and this problem is also different in which term gets the complement inside the first sum.

You have

$$\bigcup_{n=k}^\infty A_n = \bigcup_{n=k}^\infty \left(A_n\setminus A_{n+1}\right) \cup \bigcap_{n=k}^\infty A_n\tag{1}$$

for all $k$. Apply the Borel-Cantelli lemma to $B_n = A_n\setminus A_{n+1}$ and note that because of $(1)$ you have $\limsup\limits_{n\to\infty} A_n = \limsup\limits_{n\to\infty} B_n \cup \liminf\limits_{n\to\infty} A_n$. You know $P\left(\liminf\limits_{n\to\infty} A_n\right) = 0$ from $P(A_n) \to 0$.