Is my proof valid? Let $\{A_i\}_{i=0}^{i=n}$ a series of events such that $\forall i$ $P(A_i)=1$. Show that $\bigcap\limits_{0 \leq i \leq n}A_i=1$.

Question

Let $\{A_i\}_{i=0}^{i=n}$ a series of events such that $\forall i$ $P(A_i)=1$. Show that $\bigcap\limits_{0 \leq i \leq n}A_i=1$.

My attempt:

Let $0\leq k\ne j\leq n$, so $P(A_j)=1, P(A_k)=1$.

From exclusion inclusion principle:

$P(A_j \cup A_k)=P(A_j)+P(A_k)-P(A_j\cap A_k)$

$1 = 1 + 1 - P(A_j \cap A_k) \Rightarrow P(A_j\cap A_k)=1$

Could anyone confirm/be ashamed of my "work"?

Thanks in advance!

Answer 1

It ,is not enough to prove that intersection of two of the sets has probability $1$. You are asked to prove that $P(\cap_i A_i)=1$. For this note that $P(A_i^{c})=0$ for all $i$. This implies that $P(\cup_i A_i^{c}) \leq \sum_i P(A_i^{c})=0$. Hence $P(\cup_i A_i^{c})^{c})=1$. But $(\cup_i A_i^{c})^{c}=\cap_iA_i$ and this completes the proof.