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The points belonging to infinitely many $E_n$ form their limit superior of sets:For every $N$, countable subadditivity givesThus $\boxed{m(G)=0}$. Outside this null set, each point belongs to only finitely many $E_n$, so its indicator functions are eventually zero. Therefore $\boxed{{\chi }_{E_n}(x)\to 0\text{ almost everywhere}}$. This is the summable-measure argument behind the first Borel-Cantelli lemma; no independence assumption is needed.

For a sequence of sets, the limit superior of sets consists of points belonging to infinitely many members, while the lower limit consists of points belonging to every sufficiently late member. If the two sets coincide, their common value is the set-theoretic limit.