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Suppose we build a number in this way: we put the natural numbers one after the other. For example, for the first 5 numbers: $n_1=1,n_2=2,n_3=3,n_4=4,n_5=5$ we obtaine a new number $N_5=12345$. For the first 20 numbers we have in the same way: $$N_{20}=1234567891011121314151617181920$$ and so on. If we build a new number using all the natural numbers in this way, is this new element a real number? I suppose yes, because it seems to be uncountable, but I'm not sure. Can anyone suggest a proof? Thanks.

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    $\begingroup$ Your "number" would be greater than any positive integer, thus cannot be a number... $\endgroup$ – N. S. May 13 '13 at 15:14
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    $\begingroup$ The number can not be uncountable. You can just construct a countable set of digits this way. $\endgroup$ – gukoff May 13 '13 at 15:20
  • $\begingroup$ This is not [number-theory] either, but it wasn't [set-theory] too. I'm not sure how to tag this. $\endgroup$ – Asaf Karagila May 13 '13 at 15:36
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    $\begingroup$ You might be interested the read about Champernowne's constant. $\endgroup$ – MJD May 13 '13 at 15:58
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Every integer has a finite number of decimal digits, because it is a finite sum of $1$ (or $-1$ for negative integers).

While real numbers can have infinitely decimal digits, those can only appear in the fractional part. Your construction, if so, is not a real number. It's an infinite string of digits.

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