Well, what definition of "compact"

**does** your book have?

The first says that a set, A, is compact if and only if every open cover of A has a finite subcover. That is, given any collection of open sets such that every point of A is in at least one of the open sets, there exist a finite sub-collection of those same open sets such that every point of A is in at least one of the sets in the subcollection.

The second says that a set, A, is compact if and only if every infinite sequence

of points in A converges to a point in A.

To use the first, for example, since f has a finite limit at every point, x, given any

, there exist a number,

for each x such that if

then

. let

be the neighborhood about point x with radius

. These form an open cover of K. Since K is compact, there exist a finite sub-cover. Now we have a finite number of points,

such that every point in K is in some

which means that for y in

,

< f(y)< f(x)+ \epsilon[/tex]. Since the set of "x"s is finite, there is a largest such "

" and a smallest such "

" which are bounds for f(K).