In exercise 49 of Spivak's Calculus, a function $h$ is termed to be increasing at any point $a$ if there exists a $\delta > 0$ such that
$ a - \delta < x < a \implies h(x) < h(a) $ $ a < x < a - \delta \implies h(a) < h(x) $
and the reader is asked to prove that a function which is increasing at all points in some interval is increasing on that interval.
I was able to show this using Heine-Borel...
(Proof: Let $x, y \in I$, $x
$ h(x) \leq h(c_1)
so that $h(x) < h(y)$.)
...but Spivak never introduced Heine-Borel. He suggests
Prove [the result] by considering for each $b$ in $[0, 1]$ the set $S_b$ = $\{x: h(y) \geq h(b)$ $\forall y \in [b, x]\}$ (Hint: Prove that $S_b = \{x: b \leq x \leq 1\}$ by considering $\sup(S_b)$).
I admit that I don't see what he's getting at. Does somebody else know what he means?