The Borel–Cantelli lemma proof

Question

As part of the Borel–Cantelli lemma's proof we see the following:

Can you please explain me why

Am I correct in reading the inf of a prob as the smallest probability among all the for all the possible $N\geq 1$ values? Which would then tend to $0$ as $N$ tends to infinity?

Answer 1

Suppose that $A$ and $B$ are any two events. Then $$P(A\cap B)\le\min\{P(A),P(B)\}$$ since $P(A\cap B)\le P(A)$ and $P(A\cap B)\le P(B)$. In the proof, we have a countable number of events and the minimum might not exist. Hence, we take the infimum of these events and obtain $$ P\biggl(\bigcap_{N=1}^\infty\bigcup_{n=N}^\infty E_n\biggr)\le\inf_{N\ge1}P\biggl(\bigcup_{n=N}^\infty E_n\biggr). $$