Infinite subset in compact metric space
As it stands the conclusion is trivial: for a non-empty set we can always use a constant sequence with value in $A$.
The intended conclusion is more along the lines of
There is a sequence $(a_n)_n$ with all $a_n \in A$ so that all terms are distinct which is a Cauchy sequence.
For that we can indeed use the limit point compactness or ,even better, the sequential compactness of $X$.