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I'm trying to prove that $X\subset \mathbb N$ is countable, in another words, if there is a injection $i:X\to \mathbb N$ then $X$ is countable. I know this is true intuitively but I couldn't find any easy proof of this fact. My attempt is to find another injection $g:\mathbb N\to X$ and use Cantor-Bernstein theorem.

I need help to find a simple proof of $X\subset \mathbb N$ is countable.

Thanks

user42912
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1 Answers1

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When $X$ is finite, there is nothing to prove, so we assume $X$ infinite.
By the Well-Ordering Principle, $X$ contains a least element, denoted as $x_1.$ Then $X-\{x_1\}$ has a least element, denoted as $x_2,$ and so on. Then you find an injection: $\mathbb N\rightarrow X,$ sending $n$ to $x_n.$
Hope this helps.

awllower
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