In "Sur la nature arithmétique du nombre e" (Comptes rendus de l'Académie des Sciences 128 (1899), 596-9) Borel presented his result on transcendence measure for e. This can be restated as follows:
For any positive $\varepsilon$ and for any polynomial $P \in \mathbb{Z}[x]$ with degree n satisfying $|P(e)| < \varepsilon$, there exist $M = M(n) >0, k = k(n) > 0$ such that $$H \geq M\left(\dfrac{1}{\varepsilon}\right)^{\frac{k}{\ln \ln \frac{1}{\varepsilon}}},$$ where H denotes the height of P.
I wonder, how he derived this bound? It may be obvious, but I can't figure it out.
I know that there exist much better results, but this seems to be a simple consequence of Hermite's theorem.
In addition, this proposition should be equivalent with following assertion:
For any positive integer n, there are only finitely many polynomials $P \in \mathbb{Z}[x]$ with degree n and height H satisfying $|P(e)|< H^{-c \ln \ln H}$ for some $c=c(n) > 0$.
How can this equivalence be shown?