Read an Alternative Proof of the Extreme Value Theorem Using Tagged Partitions

Background

Last Saturday, I read an article on using gauges defined on a close and bounded interval $I$ (i.e. strictly postive functions over $I$) and tagged partitions to prove the Extreme Value Theorem.1 Though the full text is not free, one can preview the first page, on which an attractive introduction is found.

Motivation

In ordinary proofs of the Extreme Value Theorem, one makes use of the fact that continuous functions defined on $I$ are bounded, which is proved by the Bolzano–Weierstrass Theorem, which has to be proved by either the Monotone Convergence Theorem or the Nested Interval Theorem. A student who prepares for an exam in math analysis will revise the lengthy proofs of these theorems.

However, the alternative proof of the Extreme Value Theorem is just a proof by contradiction making use of basic properties of $\delta$-fine partitions and the fact that continuous functions defined on $I$ are bounded, which is proved by basic properties of $\delta$-fine partitions only.

Once one accepts that for any gauge $\delta$ on $I$, $I$ is $\delta$-fine, one can start reading the proof.

Problem

In the author’s proof, it’s said that if $f$ is a continuous function defined on $I$, $K:=\sup\{f(x) \mid x\in I\}$, and $\forall t\in I, \exists a(t),\delta(t)>0$ such that $|x-t|<\delta(t)$ and $x\in I$, then $f(x)<K-a(t)$.

Spending too much time on boosting the PageSpeed of the homepage of this blog, I forgot a result obtained from an elementary exercise on continuous functions, and was stuck at this point.

I tried writing inequalities for two hours, but I got nothing.

Solution

I looked at the graph, and solved the problem quickly by setting $\epsilon:=K-f(t)-a(t)$. If $|x-t|<\delta(t)$ and $x\in I$,

\begin{aligned} f(x)-f(t)\le|f(x)-f(t)|<&\epsilon=K-f(t)-a(t)\\ f(x)<&K-a(t) \end{aligned}

1. Gordon, R. A. (1998). The use of tagged partitions in elementary real analysis. American Mathematical Monthly, 107–117.