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

Proof. Let $n>|\alpha|$ We want to prove the following inequality: $$ \left(1+\frac{\alpha}{n}\right)^n \leq \left(1+\frac{\alpha}{n+1}\right)^{n+1} $$ Since both sides are positive, this follows by taking the $(n+1)$ th root and using the arithmetic-geometric-harmonic means inequality: $$ \sqrt[n+1]{\left(1+\frac{\alpha}{n}\right)^n} = \underbrace{\sqrt[n+1]{1\cdot \left(1+\frac{\alpha}{n}\right) \cdots \left(1+\frac{\alpha}{n}\right)} }_{\textrm{$n+1$ elements} } \leq \frac{1+n\left(1+\frac{\alpha}{n}\right)}{n+1} = 1+\frac{\alpha}{n+1} $$