Is there a name for the theorem that $\lim\limits_{n \to \infty} (1+\frac{1}{n})^n < \infty$?

Question

Is there a name for the theorem that $\displaystyle \lim_{n \to \infty} (1+\frac{1}{n})^n < \infty$ ? Wikipedia has a List of things named after Leonhard Euler which mentions Euler's number but not the theorem.

I wanted to write something like:

Carleman's inequality can be proved using a weak form of Stirling's inequality, which follows from $\displaystyle \lim_{n \to \infty} (1+\frac{1}{n})^n < \infty$, which can be proved using Bernstein's inequality.

but that seems awkward.

Answer 1

I don't think there's a specific theorem for this. However, noticing that $\lim_{n \to \infty}(1+\frac{1}{n})^n=\lim_{n \to \infty}e^{n\ln(1+\frac{1}{n})}$=$e^{\lim_{n \to \infty}\frac{\ln(1+n^{-1})}{n^{-1}}}$

(because $(1+\frac{1}{n})^n>0$ and $y=e^x$ is differentiable for all x)

Then, it's obvious that the expression above could be simplified with L'Hopital's rule to get $e$ (However, L'Hopital's rule is not a theorem after all...And to argue this way we have to first define $e$ as a finite constant)

Answer 2

The result $\lim\limits_{n \to \infty} (1+\frac{1}{n})^n < \infty$ is a special case of the Monotone Convergence Theorem for sequences.