Does this number belong to the set of real numbers?

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.

Answer

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.