= Solution
The event that infinitely many $A_n$ occur is $\limsup_n A_n=\bigcap_{m\ge1}\bigcup_{n\ge m}A_n$. For every $m$, the <union bound> and monotonicity of a <probability measure> give
$$
0\le\mathbb P(\limsup_n A_n)\le\mathbb P\left(\bigcup_{n\ge m}A_n\right)\le\sum_{n\ge m}\mathbb P(A_n).
$$
The right-hand side tends to zero as the tail of a convergent series. Therefore \b[the first Borel-Cantelli conclusion] is
$$
\boxed{\mathbb P(\limsup_n A_n)=0.}
$$
This proves the <Borel-Cantelli first lemma> without any <independence> assumption.
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