Map from set to its subset

Question

So I got this question:

Let $A$ be uncountable infinite set, and let $B$ be a countable subset of $A$.
Prove that $A\setminus B$ is uncountable.

So I figured that $A\setminus B\subset A$ and hence $|A\setminus B|\leq|A|$, and all that left to prove is that $|A|\leq|A\setminus B|$
but now I need to find an injective function from $A$ into $A\setminus B$ and I'm having trouble thinking about it.

Answer 1

Suppose that $A \setminus B$ were countable (i.e. not uncountable).

Then $A = A\setminus B \cup B$ would be countable too as a (disjoint) union of two countable sets. But $A$ is given to be uncountable. Contradiction.

So $A\setminus B$ is uncountable.