If $C$ is infinite and $B$ is finite, then $C\setminus B$ is infinite.

Question

If $C$ is infinite and $B$ is finite, then $C\setminus B$ is infinite.

Proof:

Suppose $A=C\setminus B$ is finite. Then since $C$ is infinite, $$C=(C\setminus B)\cup(C\cap B)=A\cup(C\cap B)$$ is also infinite. This implies three separate cases:

1. $A$ is finite and $C\cap B$ is infinite,
2. $A$ is infinite and $C\cap B$ is finite, or
3. $A$ and $C\cap B$ are infinite.

We already asssumed that $A$ was finite, so case (1) applies. Hence $C\cap B$ is infinite and so is $(C\cap B)\cup B=B$, but we assumed B was finite. This is a contradiction and so we conclude our assumption that $A$ was finite is false; $A$ must be infinite. $\square$

I was hoping someone could verify if this proof is valid or not. Thanks in advance!

Answer 1

There is no need for separate cases. Since $B$ is finite and if $A=C\setminus B$ is finite, then $$|A\cup B| = |A|+|B|< \infty$$ so $A\cup B$ is also finite. But $A\cup B = C$ and you are done.