I want to prove that a topological space is Lindelöf Well, I'm trying to prove the following exercise:
Let $X$ be a compact Hausdorff space and its cardinality is greater than the real numbers. Then $X$ is Lindelöf but not separable.
Could someone try to give an idea how to handle this problem? Thanks. :)

That $X$ is Lindelof is trivial since every compact Hausdorff space is by definition also Lindelof. So the interesting part is to prove that the cardinality of $X$ is at most the cardinality of the reals. 
The second part is false, there is a compact Hausdorff space of cardinality $2^{
|\Bbb R|}$ that is separable, namely the Stone-Cech compactification of the natural numbers. See Question about the cardinality of a space 
See also,e.g. Theorem 3.6.12 and 3.6.14 in Richard Engelking, General Topololy (1989). 
Hint. I could not think of the answer immediately, but I found the above answer when I googled:
cardinality compact Hausdorff separable space
