Could you explain how to show thm 6.3? I don't know how i should bring the concept of limit point here. I would like to know how to prove this using definitions listed below and the definition of limit point(if possible)

Theorem 6.3. If X is a compact space, then every infinite subset of X has a limit point. Definition. Let A be a subset of X and let C = {Calaen be a collection of subsets of X. Then C is a cover of A if and only if A C U, Ca. The collection C is an open cover of A if and only if C is a cover of A and each Ca is open. A subcover C' of a cover C of A is a subcollection of C whose elements form a cover of A. For instance, the open sets {(-n, n)}nen form an open cover of R. A subcover of this cover is {(-n, n)}n>5, because these sets still cover all of R. Definition. A space X is compact if and only if every open cover of X has a finite subcover