# 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.