The easiest way to prove it is by using axiom of choice. It is a bit harder to prove it without using axiom of choice.
Well a place where the axiom of choice does not hold is pretty scary. Hope you don't have nightmares ;)
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Ok, so I have come to the point where I don't see how you can prove it without assuming the axiom of choice (AC). I think I will add that to the text with a link to the axiom. And I couldn't find one.
If you can find or construct a proof that does not require AC I would very much appreciate it, as I would like to avoid mentioning AC in fear of it confusing people. The target audience of this post are non-mathematicians after all.
I don't think the proof without axiom of choice is suitable for non-mathematicians. I will put it in a post since I cannot find the proof in any literature. You can try to simplify it if you think it is worth your time.
Thank you, I'm looking forward to your post! 😃
Here is the post https://steemit.com/mathematics/@mathowl/countable-infinity-is-the-smallest-cardinal-infinity-even-without-assuming-the-axiom-of-choice-warning-technicalities-await