Proving there must exist a maximum value in a continuous interval

Question

I know this is part of the Extreme Value Theorem but I want to tackle this one bit first, focusing on the maximum case.

For a function $f$ continuous over interval $[a,b]$, it has a max value over this interval.

I define the max like so: $\exists d \in [a, b] : \forall x \in [a, b], f(x) \leq f(d)$

One way to begin tackling this is to prove by contradiction by showing the negation is false.

$\forall d \in [a, b] , \exists x \in [a, b]: f(x) > f(d)$

In other words if there is no maximum then it means for any $d$ we choose, there's a value of $x$ where $f(x)$ is even greater.

The usual proofs I see just throw a bunch of set theory and notation at the problem that I don't understand, involving things like subsequences or "compactness" for some reason. I don't understand the use, purpose, or motivation for these approaches.

Is the general idea to show that the negation implies the function going up to infinity, which somehow contradicts our assumption of continuity?

Answer 1

The negation says: "it is not bounded". So yeah the idea is to use that and reach a contradiction.

Without compactness you can't do it. Compact=bounded and closed. If it is open say on the right $[a,b)$ then the function can explode in b and still be continuous. Look at $1/x$ on $[-1,0)$.

If it is not bounded then just take literally any continuous function that goes to infinity, say a non constant polynomial...

Answer 2

There are two main hypotheses for this result: that the function is continuous, as you have noted, and that the domain which you consider is an interval, that is, a compact (bounded and closed) set.

Consider the continuous function $f(x)=x$ over $[0,1)$, that is, the set of $x$ such that $0 \leq x < 1$. You cannot find a point $d$ for which $f$ is maximal over this set : there will always be a point (closer to 1) where $f$ is larger.

The same reasoning works for the closed but unbounded set $\mathbb{R}$.

This is the reason why proving the above statement requires you to use some result relating to compactness and the fact that sequences will converge within a closed interval.