Suppose that $f_n \colon \Bbb{R}\to [0,1]$ for $n\in \Bbb{N}$ and that each one of these functions is nondecreasing. Prove that there exists a function $g\colon \Bbb{R} \to [0,1]$, a countable set $A\subset \Bbb{R}$ and a subsequence $f_{n_k}$ such that $\lim\limits_{k\to\infty} f_{n_k}(x)=g(x)$ for all $x\in \Bbb{R}\setminus A$.
I'm reviewing real analysis and came across this problem from a homework assignment: http://www.math.washington.edu/~toro/Courses/14-15/524/1-hw.pdf
(number 5, part 2)
My question is, what does nondecreasing or countable set have to do with anything? Why can't we just fix an $x$, then $f_n(x)$ is just a sequence in $[0,1]$, which is compact, hence sequentially compact, and note that it has a convergent subsequence. Then $g$ is just the function that maps to these points. In part 1 of the problem we prove that nondecreasing functions have countably many discontinuities which is very routine, so I assume the convergence doesn't happen on those points. But what goes wrong?