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).