40-01-1fracalphannIsMonotoneForLargeN.tex

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\pmtitle{$(1+\frac{\alpha}{n})^n$ is monotone for large $n$}
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\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amsthm}

\newtheorem*{lemma}{Lemma}

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

\begin{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:
\[
\sqrt[n+1]{\left(1+\frac{\alpha}{n}\right)^n} =
\underbrace{\sqrt[n+1]{1\cdot \left(1+\frac{\alpha}{n}\right)
\cdots \left(1+\frac{\alpha}{n}\right)} }_{\textrm{$n+1$ elements}
} \leq \frac{1+n\left(1+\frac{\alpha}{n}\right)}{n+1} =
1+\frac{\alpha}{n+1}
\]
\end{proof}

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