The set of point where limit function is finite Prove that the set of points at which a sequence of measurable real-valued functions converges (to a finite limit) is measurable.
Proof:
Let $f_n(x):X\to \mathbb{R}$ be a sequence of real-valued measurable functions and $\lim \limits_{n\to\infty}f_n(x)=f(x)$ for $x\in X$. We have to show that $\{x\in X: |f(x)|<\infty\}$ is measurable.
Using Cauchy criterion we get: $f_n(x)$ converges to a finite limit iff $\forall n\in \mathbb{N}$ $\exists N=N(n)$ such that $\forall k,p\geqslant N$ we have $|f_k(x)-f_p(x)|<\frac{1}{n}$.
Here we see that $N$ depends on $n$, i.e. $N:=N(n)$. And I have the following problem: wee see that
$$E=\bigcap \limits_{n=1}^{\infty}\bigcup \limits_{N=1}^{\infty}\bigcap \limits_{k,p\geqslant N}\{x: |f_k(x)-f_p(x)|<\frac{1}{n}\}.$$
$x\in \text{RHS}$ $\Rightarrow$ $\forall n\in \mathbb{N}$ we get that $x$ belongs to inner union $\Rightarrow$ $\forall n\in \mathbb{N}$ $\exists \tilde{N}\in \mathbb{N}$ such that $x$ belongs to inner intersection.
Here's one problem worries me.
How to understand that $\tilde{N}$ is the same as $N$?
Sorry if this topic is repeated but this question seems to me interesting.
 A: This is just a matter of grinding through the equivalences.
You just need to show that $f_n(x)$ is Cauchy iff $x \in E$.
If $f_n(x)$ is Cauchy, then for all $n$ there is some $N$ such that for all
$k,p \ge N$ then $|f_k(x)-f_p(x)| < {1 \over n}$.
Let $A_{n,k,p} = \{x | |f_k(x)-f_p(x)| < {1 \over n} \}$.
Then if $f_n(x)$ is Cauchy, then for all $n$ there is some $N$ such that
$x \in \cap_{k,p \ge N} A_{n,k,p}$.
Then if $f_n(x)$ is Cauchy, then for all $n$,
$x \in \cup_N \cap_{k,p \ge N} A_{n,k,p}$.
And finally, if $f_n(x)$ is Cauchy, then $x \in E= \cap_n\cup_N \cap_{k,p \ge N} A_{n,k,p}$.
Now suppose $x \in E$. Then for all $n$, we have
$x \in \cup_N \cap_{k,p \ge N} A_{n,k,p}$.
Hence for all $n$, there is some $N$ such that $x \in \cap_{k,p \ge N} A_{n,k,p}$.
And finally, for all $n$ there is some $N$ such that for all
$k,p \ge N$, $x \in A_{n,k,p}$, or equivalently
$|f_k(x)-f_p(x)| < {1 \over n}$.
A: The point is that, in examining whether two sets are equal, we are essentially examining the property of each of their elements, here $x$, but not $N$ or $\tilde N$. All we care about is: for a point $x\in E$, and for each $k$, does there always exist one natural number $N(x,1/k)$ (whose exact value we don't care) such that $|f_m(x)-f_n(x)|<1/k$ as long as $m,n>k$? If so then $E\subset \{ x\mid f_n(x)\,\text{converges}\}:=F$. Conversely, we ask, for each $x\in F$, such that $f_n(x)$ is a Cauchy sequence, does it always qualify as an element in $E$? Then we have to do some "for any/for all" arguments, but again, we don't really need to care about the exact values arising in them. 
