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?