Smallest positive real number -- history of mistake When I was about 14 (it was long ago by the way) I was sure that there is the smallest number in $\mathbb{R}_{>0}$ (i.e. a real number which is strictly greater than 0). I even somehow knew that the name of this number is $\varepsilon$. Now I know that such number does not exist, this is not the question. But recently I met a guy (he is about 14 now) and he has the same idea of smallest positive number.
I am quite sure that I am not the first one who has done this mistake. People who created analysis probably did this first.

Question 1. Are there famous mathematicians who did this mistake?
Question 2 Is there a standard name for this mistake?

Achilles and the Tortoise is similar but not exactly this.
I am in particular interested in Leibniz Monadology, but it is too vague (at least for me).

Question 3 Did Leibniz have this wrong belief?

Remark 1.  The concept of infinitesimal numbers is a less naive one. If you have an infinitesimal number $\epsilon$ then you can consider $\epsilon/2$ (why did not I think about this when I was 14...).
Remark 2.  I aware of NSA. But my question is strictly about real numbers (not hyperreal or whatever). I am not asking about people who wanted to formalize the concept.
 A: I do not know about points 2 and 3 but with respect to point 1:
You might be interested in the wikipedia-article about non-standard-analyis (NSA).
While this does not technically relate to mathematicians believing in a "smallest real number" (- I believe the giants lending us their shoulders to stand on were pretty smart, even in ancient greece... -) the concepts of NSA document that there has been a notion that $\varepsilon$-$\delta$-techniques are somewhat unsatisfactory and "true infinitesimals" should be formalized. So while likely not falling for the same mistake you are describing, mathematicians clearly have been puzzled by related problems.
A: Within Nelson's Internal Set Theory, you can work within the real number line itself.  Here you would choose a fixed nonstandard $\epsilon>0$ and work with a grid of numbers between $-H$ and $H$ where $H=\frac{1}{\epsilon}$.  This is enough to do most of calculus, including derivatives and integrals, as well as most of probability theory; see Radically Elementary Probability Theory.  This setting provides a satisfactory formalisation for the idea of a "smallest real number", namely $\epsilon$.
