If $C\subseteq [0,1]$ is uncountable then $C\cap[\alpha,1]$ is uncountable
- No need to prove anything, one counterexample suffice to prove a statement is false.
- Because for $a=1$, we have $[a, 1]=\{1\} $ countable, and for $a=0$, it's $[0,1]\cap C=C$ uncountable by hypothesis.
- However, if we take $\inf$ instead of $\sup$, the statement becomes true. Can you prove it?