TITLE: Error in a proof of reverse Fatou's lemma
QUESTION [0 upvotes]: I have the following steps while tackling reverse Fatou's lemma:
$P(\limsup A_n)=P(\cap_N \cup_{n\ge N} A_n)=\lim_{N \to \infty}P(\cup_{n \ge N} A_n)\le \limsup_{N\to \infty} P(\cup_{n \ge N} A_n)\le\limsup_{N\to \infty} \sum_{n \ge N}P(A_n)\le\limsup_{n\to \infty}P(A_n)$.
Most steps are usual ones; the first inequality is as the limit of any sequence must be lesser than the limit superior. 
The result I get eventually seems to be the opposite of reverse Fatou's lemma. Could someone explain which step above is wrong? Is there any counterexample to that step?

REPLY [1 votes]: The step $\limsup\limits_{N\to \infty} \sum\limits_{n \geqslant N}P(A_n)\leqslant\limsup\limits_{n\to \infty}P(A_n)$ is wrong.
Assume for example that $P(A_n)=1/n$ for every $n\geqslant1$, then the LHS is $+\infty$ and the RHS is $0$.
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