How Do Probability Properties Influence Event Outcomes?

[Kudasai]

Let $(\Omega,\mathcal F,P)$ be a space of probabilities and $(A_n)_n\subseteq \mathcal F.$ Show that

a) if the sequence satisfies $\inf\{P(A_n)_n:n\ge1\}=\alpha,$ with $\alpha\ge0,$ then $P\left( \bigcap\limits_{n=1}^{\infty }{\bigcup\limits_{k=n}^{\infty }{{{A}_{k}}}} \right)\ge \alpha .$

b) if the events $A_1,\ldots,A_n$ satisfy $P\left( \bigcap\limits_{i=1}^{n}{{{A}_{i}}} \right)=1$ then $A_1,\ldots,A_n$ are independent events.

c) If $A,B\in\mathcal F$ then show that $P(A\cap B)\ge P(A)+P(B)-1$

Any ideas to solve a)? For b), I should prove that $P(A_1\cap\cdots\cap A_n)=P(A_1)\cdots P(A_n),$ right? But I don't see how to use the fact I'm given.
As for c), I think I'm missing some property of sets, or any trick there. How to takle it?

Thank you.

 

[Valkarie]

a) By the Monotone Convergence Theorem, we have$$P\left( \bigcap\limits_{n=1}^{\infty }{\bigcup\limits_{k=n}^{\infty }{{{A}_{k}}}} \right)=\lim_{n\to\infty}\,P\left(\bigcup_{k=n}^{\infty }{{{A}_{k}}}\right)=\lim_{n\to\infty}\,\sum_{k=n}^{\infty}P(A_k)\ge \lim_{n\to\infty}\,\inf_{k\ge n}\,P(A_k)=\alpha.$$b) Let $A_1,\ldots,A_n\in\mathcal F$ such that $P\left( \bigcap\limits_{i=1}^{n}{{{A}_{i}}} \right)=1.$ We have$$1=P\left( \bigcap\limits_{i=1}^{n}{{{A}_{i}}} \right)=P(A_1\cap\cdots\cap A_n)=P(A_1)\cdots P(A_n).$$This shows that $A_1,\ldots,A_n$ are independent events.c) Since $A,B\in\mathcal F$, we have$$P(A\cap B)\ge P(A)+P(B)-1$$by the subadditivity of the probability measure.