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## Homework Statement

Show if K contained in R is compact, then supK and inf K both exist and are elements of K.

## Homework Equations

## The Attempt at a Solution

Ok we proved a theorem stating that if K is compact that means it is bounded and closed.

So if K is bounded that means that for every element k in K there exists an M in R so that,

|k|<M

which is equiv to:

-M<k<M

meaning for all k in K

k>-M and k<M and therefore K is bounded above AND below.

According to the axiom of completeness, that means that the supK and the infK exists.

Alright, this is where I get stuck, I know that the supK and infK are the limit points of K, and since we know that K is closed, they would be contained in K, but I don't know how to show that supK and infK are the limit points. Any help would be great!