How to prove this problem ($g(x)$ approximated by nondecreasing $f_n(x)$)

Question

How to show the following problem:

My approach is following:

1. From the dexcription of the problem, $g(x)$ should be non-decreasing.
2. By construction, let $g(x)=\text{sup}_{r\leq x} g(r)$ to make sure $g(x)$ is non-decreasing.
3. From the knowledge that non-decreasing function has countable discontinuities.

Then, how to do the following steps?