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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?

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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.

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  • $\begingroup$ Oops, it was supposed to be a less than sign! $\endgroup$ – Ldog327 Apr 21 '15 at 18:58
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    $\begingroup$ So, then it seems like the set is clearly finite and only contains {0, 1, 2, 3} ? $\endgroup$ – Ldog327 Apr 21 '15 at 18:58
  • $\begingroup$ Hm, check again, I hope the signs are now right and the ones you have in your actual problem statement? $\endgroup$ – Hagen von Eitzen Apr 21 '15 at 20:16

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