I have been reading the book at http://www.neunhaeuserer.de/short.pdf, and have noticed that in the proof of the intermediate value theorem (Theorem 5.8 in the book), it seems to be quietly assumed that you can always find a sequence of points in a set that tend to the supremum of the set (in $\mathbb{R}$).
I presume this is true, but how would one go about proving it? I can't seem to find the result easily, but maybe I'm searching the wrong. What does one call this property (of there being a sequence converging to you?). It seems "limit point" would be the natural term, but I've discovered that the definition doesn't quite mean that--e.g. an isolated point $a$ cannot be limit point, but the sequence $a,a,a,a,a,a \dots$ converges to it (or is that a matter of how we define convergence?).