Counter example to Borel Cantelli Lemma when assuming only that probabilities converge to zero

Question

As I understand it, the (First) Borel Cantelli Lemma says that if $$\sum_{n=1}^\infty P\{E_n\} < \infty$$ then $$P\left\{\bigcap_{m=1}^\infty \bigcup_{n=m}^\infty E_n\right\} = 0.$$ Why is it not sufficient that $$P\{E_n\} \to 0 $$ can anyone provide a counterexample?

Answer 1

Suppose the $E_n$ are independent with $P(E_n)=\frac1n \to 0$ but $\sum\limits_{n=1}^\infty P(E_n) = \infty$

Then $P\left(\bigcap\limits_{m=1}^\infty \bigcup\limits_{n=m}^\infty E_n\right) = 1$ as you expect an infinite number to occur