# Thread: Trouble with Quick Proof

1. ## Trouble with Quick Proof

Let A and B be sets.

Prove that if A is infinite and A C B, then B is infinite.

My guess would be:

A is infinite so A=/= the empty set and A is not equivalent to Nk for some natural number k.

So obviously I know that since B is bigger than A, then B is clearly infinite, but I really don't know how to show this.

2. Originally Posted by zhupolongjoe
Let A and B be sets.
Prove that if A is infinite and A C B, then B is infinite.
My guess would be:
A is infinite so A=/= the empty set and A is not equivalent to Nk for some natural number k.
It appears that your text material use the definition, “A is infinite if A is not finite”.
If B were finite, what does that imply about A?

3. That is indeed the definition in the text.

If B were finite, then A would be finite

I suppose it should be a proof by contrapositive?

I.e. Assume B is finite.

Then since A C B, A is finite, and so the contrapositive is true.

Is it that simple?

4. Originally Posted by zhupolongjoe
Assume B is finite.
Then since A C B, A is finite, and so the contrapositive is true.
Is it that simple?
Indeed it is.