# $P(\bigcup_{i=1}^{\infty}A_i)\leq \sum_{i=1}^{\infty}P(A_i)$ when $A_1,A_2,… \in \mathcal{A}$ [duplicate]

Possible Duplicate:
How to prove Boole’s inequality

The set of events $\mathcal{A}$ is an collection subsets of $\Omega$ where:

D1: $\Omega \in \mathcal{A}$

D2: $A\in\mathcal{A}\implies A^c\in\mathcal{A}$

D3: $A_1,A_2,...\in\mathcal{A}\implies\bigcup_{i=1}^{\infty}A_i\in\mathcal{A}$

The probability measure $P:A\to\mathbb{R}$ is an image from $\mathcal{A}$ to $\mathbb{R}$ where:

(D4) $\forall A\in\mathcal{A},(0\leq P(A)\leq 1)$

(D5) $P(\Omega)=1$