If A and B are denumerable, then A-B is denumerable I am studying for finals and I came across this question:

If $A$ and $B$ are denumerable, then $A-B$ is denumerable.

I saw this had been asked in the past but there was not a concrete answer as to whether this can be proved or not.

I would think $A-B$ would still be denumerable but not sure since denumerable means the set is equivalent to the natural numbers.
Lemma:
A subset of a countable (denumerable) set is countable.

Proof of lemma: Let $A$ be a countable set and $B\subseteq A$. Since A is countable there is an injection $f:A\to \mathbb{Z^+}$. Now the restriction of $f$ to $B$ is an injection from $B$ to $\mathbb{Z^+}$. Hence $B$ is countable. $\Box$

Now it is required to prove that $A-B$ is countable (denumerable). Obviously $A-B\subseteq A$. Since $A$ is countable, by lemma, it follows that $A-B$ is countable. $\Box$
Lemma:
A subset of a countable (denumerable) set is countable.

Proof of lemma: Let $A$ be a countable set and $B\subseteq A$. Since A is countable there is an injection $f:A\to \mathbb{Z^+}$. Now the restriction of $f$ to $B$ is an injection from $B$ to $\mathbb{Z^+}$. Hence $B$ is countable. $\Box$

Now it is required to prove that $A-B$ is countable (denumerable). Obviously $A-B\subseteq A$. Since $A$ is countable, by lemma, it follows that $A-B$ is countable. $\Box$