Proof of the Stable Matching Problem
The claim is not that every woman ends with the last man on her list. Rather, consider all stable matchings, and all partners of some woman $w$ in these stable matchings. Among them, pick the worst one (according to her view) $m$. Then in the man-optimal matching, $w$ is matched to $m$.

Now that you understand what they're trying to prove, read the proof again. (It couldn't have made sense before, because the claim as you stated it just isn't true.)