Why $E$ should be written like this?

Question

The question is given below and a part of its answer:

But I do not understand why $E$ should be written like this? Could anyone explain this for me please?

Answer 1

We have $x\in E$ iff $\{f_n(x)\}$ is Cauchy, that is, iff $$ \forall\epsilon>0\colon \exists m\colon \forall i\ge m\colon\forall j\ge m\colon|f_i(x)-f_j(x)|<\epsilon$$ Clearly, it suffices to consider only $\epsilon$ of the form $\frac 1n$, i.e., $$\begin{align}x\in E&\iff \forall n\colon \exists m\colon \forall i\ge m\colon\forall j\ge m\colon|f_i(x)-f_j(x)|<\frac 1n\\ &\iff \forall n\colon \exists m\colon \forall i\ge m\colon\forall j\ge m\colon x\in\{\,y\in\Bbb R\mid |f_i(y)-f_j(y)|<\tfrac 1n\,\}\\ &\iff \forall n\colon \exists m\colon \forall i\ge m\colon x\in\bigcap_{j=m}^\infty\{\,y\in\Bbb R\mid |f_i(y)-f_j(y)|<\tfrac 1n\,\}\\ &\iff \forall n\colon \exists m\colon x\in\bigcap_{i=m}^\infty\bigcap_{j=m}^\infty\{\,y\in\Bbb R\mid |f_i(y)-f_j(y)|<\tfrac 1n\,\}\\ &\iff \forall n\colon x\in\bigcup_{m=1}^\infty\bigcap_{i=m}^\infty\bigcap_{j=m}^\infty\{\,y\in\Bbb R\mid |f_i(y)-f_j(y)|<\tfrac 1n\,\}\\ &\iff x\in\bigcap_{n=1}^\infty\bigcup_{m=1}^\infty\bigcap_{i=m}^\infty\bigcap_{j=m}^\infty\{\,y\in\Bbb R\mid |f_i(y)-f_j(y)|<\tfrac 1n\,\}\\ \end{align}$$

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Note however that it is not fully trivial that the individual $\{\,y\in\Bbb R\mid |f_i(y)-f_j(y)|<\tfrac 1n\,\}$ are measurable - this may depend on which results you already have available. Thus it might be more advisable to write $$ E=\bigcap_{n=1}^\infty\bigcup_{a,b\in\Bbb Q\atop a<b<a+\frac1n}\bigcup_{m=1}^\infty\bigcap_{i=m}^\infty f_i^{-1}([a,b])$$ Do you see why that would be allowed?