Error in a proof of reverse Fatou's lemma

I have the adhering to actions while taking on reverse Fatou is lemma:

$P(\limsup A_n)=P(\cap_N \cup_{n\ge N} A_n)=\lim_{N \to \infty}P(\cup_{n \ge N} A_n)\le \limsup_{N\to \infty} P(\cup_{n \ge N} A_n)\le\limsup_{N\to \infty} \sum_{n \ge N}P(A_n)\le\limsup_{n\to \infty}P(A_n)$.

The majority of actions are common ones ; the first inequality is as the restriction of any kind of series have to be minimal than the restriction premium.

The outcome I get at some point appears to be the reverse of reverse Fatou is lemma. Could a person clarify which action above is incorrect? Exists any kind of counterexample to that action?

2022-07-25 20:46:18
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