$X$ is an arbitrary topological space which is compact . We have to prove that $X$ is limit point compact .
So , $A$ be any infinite set if $X$ . If we suppose $A$ "has no limit points" then only the points inside $A$ can be considered as limits of $A$. Thus $A$ becomes a closed set . Now for each $a\in A$ , we choose a nbd $U_a$ of $a$ such that $U_a$ intersect $A%$ at only one point $a$ .
Now here is the problem : How do I make sure that such a nbd $U_a$ exist $?$ Thnks.
$A$ has "no limit points" implying all points of $A$ are isolated points. That is taken any point of $A$ , $a$ a nbd like $U_a$ must exist for otherwise that point $a$ will become a "limit point" of $A$ .