$o$ denotes infinitesimal
In classic analysis $o\notin \mathbb R$ and $\infty\notin\mathbb R$
Because the completeness of real number, there are no gaps between real numbers. Denote the real number right after $0$ as $r$, then:
Is $1/r\in \mathbb R$?
Is $r/2\in \mathbb R$?
First, an issue at the beginning. I don't know what your "$o$" represents. If it's supposed to denote "$0$" (zero), then you are incorrect: $0$ is a real number. EDIT: You clarify that $o$ is an infinitesimal. In that case, what you've written is correct - no infinitesimal is a real number - although it's slightly misleading, since it suggests that there is some specific infinitesimal called "$o$." In fact, even in the infinitesimals as usually construed, there is no "smallest infinitesimal" after zero: we can always divide $o$ by $2$, as with real numbers.
More to the point, this question is founded on an incorrect assumption. When you write " Denote the real number right after $0$ as $r$," you are assuming that there is such a real number. (Think about if I said "Denote the biggest real number as $s$" - that'd be wrong of me, since there isn't a biggest real number!)
In fact, you're close to the proof that there is no such number: from the basic axioms for the real numbers, we can show that if $x$ is a real number then so is ${x\over 2}$, and if $x>0$ then $0<{x\over 2}<x$. (This is why you're finding it confusing to think about ${r\over 2}$ - it's clear that that should be between $0$ and $r$, despite what you've assumed $r$ is!) So there is no "real number right after $0$" (or right after anything else, for that matter). Indeed, this is (part of) what is meant by " there are no gaps between real numbers."