The answer is: No, endless division does not lead to nothing. It leads to "not nothing." This is one answer, anyway, to this a wonderful old question, going back at least to Zeno and "resolved," for practical purposes at least, by the "limit" in Calculus.
The uneasy relations between "zero" and "infinity" led the Greeks to abhor and avoid both. They are unobservables that simply bring in endless problems of incoherence and the ever-lurking "infinite regress," the quicksand of dialectic. By contrast, the atom of Democritus was, unlike ours, definitively "something" bordered by definitive "void."
What you get at the "very bottom" of infinite divisibility became both a mathematical and theological debate when Newton and Leibniz formalized the "infinitesimal" in calculus. Was the least possible "something" the same as "nothing"? All of physics depends on the answer being... no. It is a "not nothing," which Berkeley memorably characterized as "the ghosts of departed quantities."
In physics, Wheeler addressed the problem by coining the phrase "it from bit." This means that "information" is the fundamental unit, and is minimally defined as a "bit" or a "something/nothing" an "either/or" a "(0/1)" And it is precisely because of this irreducible instability or "possibility" that we have "something" and motion, rather than just "nothing."
This is, to the horror of hard-headed physicists, not unlike what Hegel suggests at the beginning of his Logic by deriving "becoming" from the irreducible overlap and interplay of the meanings of "being" and "nothing." To say something "is" is to say "nothing" about it. Thus, the idea of "being" itself gives rise to the idea of "nothing." Which now itself has a "being." And because these bare ideas collapse into one another they give rise to "becoming"... or transformation, motion, change.
So, the problem is not that such knowledge is "impenetrable," but that it is all too penetrable. One arrives by analytic divisibility not at the "atom" or at "nothing," but at an irreducible instability. The answer is that infinite divisibility leads to something that is "not nothing" and is also the generative power of "nothingness" or "negation." Which Sartre, incidentally, equates with us. For after all, there is always "something else" which is doing this endless dividing.