Intersection of uncountable sets. Let $C \subseteq  [0,1]$ be uncountable. Let $A$ be the set of all $a \in (0, 1) $ such that $ C \cap [a,1] $ is uncountable and set $\alpha = \sup A$. Is $C \cap [\alpha, 1]$ an uncountable set?
I know that $a$ must exist, but I'm not sure how to go about proving/disproving this statement. 

Take $C=[0,1]$. Then $A= (0,1)$, hence $\sup(A)=1$. Then $C \cap [\sup(A),1]= \left\{ 1 \right \}$, which is countable.
