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Full disclosure: I am a total layman at math, nothing in my training aside from high school courses and one college calculus class. I'm sure a week doesn't pass without someone posting a question about or challenge to Cantor. I am not here to challenge anything but rather to learn where the error is in my thinking. I hope I don't get mistaken for a crank who's trying to disprove Cantor.

Like many people, I suppose, I'm having difficulty wrapping my head around Cantor's proof that the real numbers are uncountable and that there are different sizes of infinity. I can follow along with the diagonal proof for the most part, but another way of visualizing the problem keeps making me think that I don't understand how the real numbers diverge from other numbers in countability.

Here is what I mean. I think it is true that any given real number is made up of a countable set of digits, following this general pattern or mask:

...NNN.NNN...

where N is any digit 0-9. To the left of the decimal point, one could imagine an endless string of 0s leading up to the actual number (if it is non-zero). Such as ...000003.14159...

The set of those digits is countably infinite, as I understand it, because it can be put into a 1:1 correspondence with the natural numbers. If that is wrong, I'd be grateful for an explanation why.

Now - and this is where I get confused - I believe I have read that the union of two countably infinite sets is also countable. I got that from some Google searches, so I have no idea if that is really true. Whether it is or not affects whether I can even get to the next step of my thinking.

If it is true that the union of countably infinite sets is also countable, then it seems to me that one could simply do another union with the next countably infinite set of digits for another real number, and so on. Even if in principle there are real numbers that cannot be listed as has been proved by the diagonal argument, the real numbers are out there, each one consisting of a countably infinite set of digits, so I don't understand how their "total" could comprise an uncountably infinite set.

How could such an ongoing sequence of unions produce an uncountably infinite set? Is there even a proof or explanation that a layman like me could comprehend?

Thanks for any help.