proving infinite set from a finite and infinite set

Question

My question reads:

If A is finite and B is infinite, then B-A is infinite.

Answer 1

Here is a way to prove it directly.

If $B$ is infinite, there is a injective function $\beta:\mathbb{N} \to B$.

Since $A$ is finite, there is some $N$ such that for $n > N$, $\beta(n) \notin A$.

Then the function $\beta'$ defined by $\beta'(n) = \beta(n+N)$ is an injective function $\beta':\mathbb{N} \to B\setminus A$, and so $B\setminus A$ is infinite.