Here is his proof found in his book Pugh's Real Mathematical Analysis:
In his second case where he says "By continuity at $c$ ...", I don't understand why bringing up the definition of continuity is significant here. If $c$ is the least upper bound of $X$, wouldn't it already imply that l.u.b. $V_{t}$ is less than $M$ since $c\in X$? Is that where I went wrong, in that $c$ does not actually belong in $X$ (well we know this is eventually true in the sentence right after but I mean based off the l.u.b. alone)? Edit: looks like I can't actually conclude that it belongs in X immediately from that fact. but I'm not exactly sure why it is not enough from the fact that $c$ is the l.u.b. of $X$ and what role does the definition of continuity play in solidifying that l.u.b. $V_{t}$ $<M$?