Smallest next real number after an integer This might be a silly question, but is it possible at all for n.00000...[infinite zeros]...1 to be the next real number after n? If not, why not?
Firstly, I know (I think) that $$\lim_{x\to \infty} \frac{1}{10^x} = 0$$ but I'm not talking about taking its limit. Surely I'm not required to.
The obvious rebuttal is that n.00000...[infinite zeros]...1 (let's just call this number c) is that c divided by two will be between n and c, which is obviously a closer number to n than c. But why is c necessarily divisible? There are special exceptions for other numbers. For instance, if b=0 in $\frac ab$, we say it's undefined.  Why can't $\frac c2$ be similarly undefined? Or something similar to the notion that infinity divided by two is still infinity?
OR, does c equal n.0, similar to how 0.999... = 1?
Apologies if this question has been asked a million times before (I was not able to find it asked quite this way) or if you find it stupid.

A simple proof that may or may not work:

.999... = 1
.999... + c = 1
c = 0
n + c = n + 0
n + c = n

Just an idea that is, at the very least, easy to understand.
