Consider the theorem for the continuous function:
Let $a be real numbers, and let $f:[a,b]\to{\bf R}$ be a function continuous on $[a,b]$. Then $f$ is a bounded function.
The proof in the classical textbook on real analysis uses the Heine-Borel theorem. It dose not say how to find the bound for $f$, but it show that having $f$ unbounded leads to a contradiction.
Here are my questions:
Is there a
direct[EDITED: constructive] proof for this theorem?More generally, can a theorem in mathematics always have a constructive proof? Or what kind of statements do not have any constructive proof, say, one has to use techniques such as "proof by contradiction" in order to prove it?