I do not understand why $E_n$ is measurable in this proof

Question

Suppose we have a sequence of measurable functions $\{f_n\}$ on $X$. Also, suppose that

(a) $0\leq f_1(x)\leq f_2(x)\leq ...\leq\infty$ for every $x\in X$

(b) $f_n(x)\rightarrow f(x)$ as $n\rightarrow\infty$ for every $x\in X$

Now, let $s$ be any simple measurable function such that $0\leq s\leq f$ and let $c$ be a constant such that $0<c<1$, and define $$E_n=\{x:f_n(x)\geq cs(x)\}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(n=1,2,3,...)$$ According to the proof, each $E_n$ is measurable. I don't understand why. Could someone help understand as to why it is measurable.

Answer 1

It is the preimage of $[0,\infty)$, a measurable set, for a measurable function $f_n(x)-cs(x)$.