prove if a A is uncountable and A is a subset of B, then B is uncountable
Since $\displaystyle A$ is uncountable it means $\displaystyle |A| > \aleph_0$. Since $\displaystyle A\subseteq B$ it means $\displaystyle |A|\leq |B|$ because the identity map on $\displaystyle A$ will be an injective function. Thus, $\displaystyle \aleph_0 < |A| \leq |B|$.