# Note on the Extreme Value Theorem

The *Extreme Value Theorem* states that a continuous function from a *compact* set to the real numbers takes on minimal and maximal values on the *compact* set. (A compact subset of n-dimensional Euclidean space may be taken as any set that is *closed* (contains the limits of all convergent sequences made of points from the set) and *bounded* (contained within some finite n-dimensional "box").

We first prove the *Bounded Value Theorem* - the range of a continuous function on a compact set is bounded. Suppose not. Now proceed by *successive bisection*: bisect the original compact set (here is where we use the boundedness); on at least one piece, the function is unbounded. Bisect that piece again. (If the function is unbounded on both pieces, pick either one). Proceeding in this way, we obtain a nested sequence of boxes, of arbitrarily small maximum dimension, converging to a single point, say, *c*, in the original set (here is where we use the closedness). Since the function is continuous at *c*, there is a box, say B, containing *c*, such that for points within the box, all the function values differ by, say, no more than 1 from the value of the function at *c* - in other words, the function is bounded on the box B. But we claim to have produced a sequence of boxes around *c* of arbitrarily small size (some of which must necessarily fit entirely inside the box B) on which the function is *un*bounded. This contradiction proves the Bounded Value Theorem

To prove the *Extreme Value Theorem*, suppose a continuous function *f* does not achieve a maximum value on a compact set. Since the function is bounded, there is a least
upper bound, say M, for the range of the function. Consider the function *g* = 1/(*f* - M).*f* never attains the value M, *g* is continuous, and is therefore itself bounded. That implies that *f* does not
get arbitrarily close to M, in contradiction to the choice of M as the *least* upper bound of the range of *f*. The same proof applies to the minimum value of *f*.

|Contact| |Front page| |Contents| |Algebra|

Copyright © 1996-2018 Alexander Bogomolny