Let , $f_n:E\to [0,\infty]$ be a monotone increasing sequence of measurable functions converges pointwise to $f$ ; where $E$ is a measurable set. Choose a simple function $\phi:E\to [0,\infty)$ such that $\phi \le f$ on $E$ and choose a real number $a(0<a<1)$. Let , $E_k=\{x\in E:f_k(x) \ge a\phi(x)\}$ , for all $k\in \mathbb N$.
We have to prove , $\displaystyle \bigcup_{k=1}^{\infty} E_k=E$.
It is a part of the proof a theorem and my book says it is clearly $\displaystyle \bigcup_{k=1}^{\infty} E_k=E$. But I'm unable to understand it.
My Work:
From definition of $E_k$ it is clear that $E_k\subset E$ for all $k\in \mathbb N$. Then , $\displaystyle \bigcup_{k=1}^{\infty} E_k\subset E$. Also it is clear to me that $E_1\subset E_2\subset E_3 \subset \cdots$ , as $f_k\le f_{k+1}$. But how I prove the reverse inclusion ? Or any other way to show the equality.