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.
The set of real numbers are (usually) defined in a way that has nothing to do with decimal representations -- they are defined by their arithmetic and geometric properties. e.g. among other things, if $a$ and $b$ are distinct real numbers, then $(a+b)/2$ is a real number that is between them.



The set of decimals are defined as being sequences of digits: there is one place for every integer. e.g. $0$ corresponds to the one's place, $1$ corresponds to the ten's place, $2$ to the hundred's place, $-1$ to the tenth's place, and so forth. Each place gets a single digit (0 through 9) assigned to it. When we write a decimal like


  123.45


we implicitly mean that all of the remaining positions get filled with zeroes. i.e. in the above numeral, the thousands place contains a zero.

The key point is that each place corresponds to an integer: there aren't any other places. If we write $n.\overline{0}$, meaning that the $0$ to the right of the decimal place should be repeated infinitely, this means that we have written a $0$ in every place corresponding to a negative integer. There aren't any places remaining to the right of the decimal place to insert a $1$! So the notation $n.\overline{0}1$ makes no sense if we try to interpret it as expressing a decimal number.



We could define other sorts of radix notation that extend decimals to have additional places to the right of the decimal place, but then we have to figure out what to do with such things.

The ordinary decimals are useful because we have a way to interpret any decimal number that only has finitely many nonzero digits to the left of the decimal place as a real number. And we also have rules for doing arithmetic with them. There are some ambiguities -- e.g. does $1.\overline{0} + 0.\overline{9}$ add up to $1.\overline{9}$ because there are no carries? Or does it add up to $2.\overline{0}$ because there is a carry in every place to the right of the decimal point? -- but these ambiguities are okay because we are interpreting both possibilities as being the same real number.

But if we extend the decimals, we no longer have the ability to relate them to real numbers. And if we want to do arithmetic with such things, we're going to have to do a lot of work to define the arithmetic operations and figure out if they have any of the familiar algebraic properties we're used to and so forth.

We can construct algebras in this way in which every number has a "next" number, but such things are going to have very little to do with real numbers.
Just look at this in the first place: the reason why probably, .9999... = 1, no rounding!
{Perhaps one will argue about '0.9999...' being just a teensy amount less than 1. Well, but isn’t '.9999…' just 3 times '.3333…'? Everyone will agree with this one. And isn’t '.3333…' the same as 1/3? Yes, of course. And isn’t 3 times 1/3 equal to 1?} 

Back to our problem: can n.00000...[infinite zeros]...1 be the next real number after n?
Impossible! When you see a number like '0.9999...' (i.e. a decimal number with an infinite series of 9s), there is no end to the number of 9s. Therefore, you cannot say "but what happens if it ends in an 8?", because it simply does not end. (This is also why 0.9999... equals 1). Similarly, 'AAAA...', an infinite series of "A"s followed by a "B" would NEVER have a "B", not in actual Mathematics, not in physical Mathematics. That is why you can never reach the end of all numbers by listing them. You can try out this! 
But this is possible only in potential Mathematics, Mathematics full of imaginations, like the world has an end; time has a beginning and an end, e.t.c, potential Mathematics, that is it.