A crude approximation of the spacing of energy levels $E_i$ of complex nuclei (like uranium) is that the energy levels appear independently, with known average spacing $D$.
I'm trying to understand a derivation of the spacing distribution of these energy levels, as done on page 11 of the book Mehta, M.L. Random matrices, Third edition, 2004.
The probability that any $E_i$ will fall between $E$ and $E + dE$ is independent of $E$ and is simply $\rho dE$, with $\rho = D^{-1}$ the average number of levels in a unit interval of energy.
Let us determine the probability of a spacing $S$, that is, given a level at $E$, what is the probability of having no level in the interval $(E, E + S)$ and one level in interval $(E + S, E + S + dS)$?
For this, we divide the interval $S$ in $m$ equal parts. The levels are independently spaced, so the probability of having no levels in the interval $(E, E + S)$ is the product the probabilities of having no level in any of these $m$ parts.
If $m$ is large, so $S/m$ is small, we can write this as $(1 - \rho S/m)^m$
Now, taking the limit
$$ \lim_{m \rightarrow \infty} (1 - \rho S/m)^m = e^{-\rho S} $$
The probability of having a level in $dS$ at $E + S$ is $\rho dS$, so the probability of having no level in $(E, E + S)$ and one level in $(E + S, E + S + dS)$ is
$$ e^{-\rho S}\rho dS $$
The part that I don't understand is that the probability of having an energy level in a small region is proportional to the length of that region. This is obviously not true for large regions, since if it would be, we could say that the chance of having no level in $S$ would be $1 - \rho S$, which is only a first order approximation to the true answer (which is actually a poisson probability for having no events in an interval.) Why is it only true for small regions and what approximation is being made here?