Question about proof of Heine-Cantor (i.e. compact and continuous implies uniform continuous) If anyone has seen the wikipedia page for the Heine-Cantor theorem, I find something off about the proof it presents. It would be incredibly tedious to write it all out here because it's pretty involved, but you can find it at here. At a certain point in the proof, after $\delta$ has been defined, the author goes on to say that for all $x,y$ in the domain $M$, $d(x,y)<\delta$. Can anyone corroborate this?

That's not what it says: suppose that $d_X(x,y) < \delta$ for two arbitrary points of $X$. It might have been less confusing for you if the proof had said: "let $x,y$ be two arbitrary points of $X$ such that $d_X(x,y) < \delta$" instead:

Reason: we want to show

$$\forall x,y \in X: d_X(x,y) < \delta \implies d_Y(f(x),f(y)) < \varepsilon$$

so we start by assuming we have two points in $X$ that are distance $< \delta$ apart, and these must be _arbitrary points_ with that property, it doesn't mean that all points of $X$ are $< \delta$ apart, but if we happen to have any two that are, we must show that the distance of their images is $< \varepsilon$.