B is a countable subset of $A$. Here's what I have: $A-B$ with elements that are in $A$ but not in $B$.
$A$ is uncountable, so infinite. Take $k$ in $B$ as the largest element in finite $B$. $k$ is also in $A$, since $B$ is a subset of $A$. $k+1$ is in $A$ and not in $B$. $k+2$ is also in $A$ since $A$ in infinite, hence $A-B$ is uncountable.
Does this make sense? I am new to proofs and this made sense to me. We are learning about countable/uncountable sets so maybe I need to use different terminology? Any help would be greatly appreciated :)
The proof is wrong. The statement is also wrong. Counterexample is $A=\mathbb{R}$ and $B=\mathbb{Q}$.