This is a well known result in mathematics, but it's my first time attempting to prove it. I'm following the second book of Analysis from Folland. Below are the notations used and the theorem, from the text.
Now we have the following exercise, which is supposed to led us to prove the part b of this theorem.
I managed to prove the part a of this exercise, but part b is hard to do. The best idea I have so far is to write for each $x\in[a,b]$, $$\lim_{\delta\to 0}\sup_{|x-y|\leq\delta}f(y) = \lim_{n\to\infty}\sup_{|x-y|\leq 1/n}f(y)$$
and note that $\sup_{|x-y|\leq 1/n}f(y) = M_i$ for some convenient partition chosen. In fact, for each $n\in\mathbb{N}$ we may choose the partition $P_n$ so it there is always a $M_i$ equal to $\sup_{|x-y|\leq 1/n}f(y)$. The problem is that there is uncountable sup's for $H$ while there is countable for $G_{P_n}$. So I don't know how this approach will (if it will) work. I need some help here, thanks.