You can prove this by contradiction: if both and are countable, then is countable.
If S is an uncountable set and T is a countable subset of S. Show that cardinality of |S\T|= |S|.
S\T is a subset of S, so |S\T|<= |S|.
I was going to use the cantor-schroder-bernstein theorem, but i dont know how to proof that |S\T|=> |S|.
If there is other ways to proof this, please post it.
Any help would be nice.
Thanks in advance