# Proving a set to be countably infinite.

I'm asked to decide and prove whether the set $\{\,x\in\mathbb{N}: |x-7|>|x|\,\}$ is finite, infinitely countable, or uncountable.

I'm pretty certain it is infinitely countable. I say that since the set is a subset of $\mathbb{N}$, and $\mathbb{N}$ is countable, that the set is countable. Now, I need to prove it to be infinite. I'm having trouble. Would the best way to go about it be to suppose that it's finite, then find a contradiction?

If $x\ge 7$ then $|x-7|=x-7$, and of course $|x|=x$. Hence any $x\in \mathbb N$ with $x\ge 7$ is certainly not in your set ($x-7\not>x$), making it finite.