Intuition behind the proof that pointwise limit of measurable function is measurable.

Question

I am self-studying measure theory.There is a very important theorem in measure theory which says that a pointwise limit of measurable functions is measurable.Now,I understand the proof but do not get the exact idea behind the construction in the proof.The objective is to show $f^{-1}(a,\infty)$ is measurable for all $a\in \mathbb R$.Now in the proof they just write: $$\{x\in X\mid f(x)>a\}=\bigcup_{m=1}^\infty\bigcup_{N=1}^\infty \bigcap_{n=N}^\infty \left\lbrace x\in X :f_n(x)>a+\frac{1}{m}\right\rbrace$$.I could easily see that the equality holds but I do not get the exact idea which led to such a strange choice.Can someone help me to understand the proof more intuitively.Can it be understood visually?