# Prove that for an increasing series of events, Event N is equal to the union of events 1 to N.

Question:

Given an infinite increasing sequence of events $$E_1 \subset E_2 \subset ... \subset E_n ....$$ show that $$\bigcup_{k=1}^n E_k = E_n$$ to verify “=”, you need to use the argument “A = B if A ⊂ B and B ⊂ A"

My Attempt:

Obviously $$E_n \subset \bigcup_{k=1}^n E_k$$ as when expanded we can see that $$E_n$$ is a part of $$\bigcup_{k=1}^n E_k$$

However I am stuck on showing that $$\bigcup_{k=1}^n E_k \subset E_n$$.

My current proof involves the fact that since each event ($$E_k$$) includes its previous events ($$E_1$$ to $$E_{n-1}$$) the union of a event and its previous events is just the event($$E_k \cup E_{k-1} = E_k$$ ). However, this is effectively the statement I am trying to prove leaving me stuck.

Any help would be appreciated, thank you.

• If $x\in\bigcup_{k=1}^n E_k$ then for some $1\leqslant j\leqslant n$, $x\in E_j\subset E_n$. Sep 30, 2023 at 21:15
• Better phrased as a statement about sets. Using "Events" implies there is something else from probability theory at play at first look. Sep 30, 2023 at 22:26

If $$x$$ is in your union, by definition $$x$$ belongs to some $$E_k$$ with $$k\le n$$. But any such $$E_k$$ is a subset of $$E_n$$. That is all there is to it.