${(1+\frac{\alpha }{n})}^n$ is monotone for large $n$

Lemma.

Let $\alpha$ be a real number. The sequence${\mathrm{(}\mathrm{1}\mathrm{+}\frac{\alpha }{n}\mathrm{)}}^n$ is monotone increasing for all $n\mathrm{>}\mathrm{|}\alpha \mathrm{|}$.

Proof.

Let $n>|\alpha |$. We want to prove the following inequality:

$${\left(1+\frac{\alpha }{n}\right)}^n\le {\left(1+\frac{\alpha }{n+1}\right)}^{n+1}$$

Since both sides are positive, this follows by taking the$(n+1)$-th root and using the arithmetic-geometric-harmonic meansinequality:

$$\sqrt[n+1]{{\left(1+\frac{\alpha }{n}\right)}^n}=\underset{n+1\text{ elements}}{\underbrace{\sqrt[n+1]{1\cdot \left(1+\frac{\alpha }{n}\right)\mathrm{\cdots }\left(1+\frac{\alpha }{n}\right)}}}\le \frac{1+n\left(1+\frac{\alpha }{n}\right)}{n+1}=1+\frac{\alpha }{n+1}$$

∎