TITLE: Where does one even need Bernoulli's Inequality?
QUESTION [13 upvotes]: Which Theorems/Lemmas/Results actually use Bernoulli's inequality? I don't seem to remember using it very often - which probably makes sense, as it's not a very strong inequality and can be proven easily.
However, where do you actually use Bernoulli?

REPLY [2 votes]: We can use Bernoulli's inequality to prove that the sequence
$$
a_n=\Bigl(1+\frac1n\Bigr)^n
$$
converges as $n\to\infty$. Denote $b_n=a_n(1+n^{-1})$. We show that $b_n$ is a decreasing sequence. We have that
\begin{align*}
\frac{b_n}{b_{n-1}}
&=\frac{(1+\frac1n)^{n+1}}{(1+\frac1{n-1})^n}
=\frac{(n^2-1)^n(n+1)}{n^{2n}n}\\
&=\frac{1+\frac1n}{(1+\frac1{n^2-1})^n}
\le\frac{1+\frac1n}{1+\frac n{n^2-1}}\\
&<\frac{1+\frac1n}{1+\frac n{n^2}}
=1.
\end{align*}
Hence, $b_{n-1}>b_n$. Since $b_n\ge1$, $b_n$ converges as $n\to\infty$ which in turn implies that $a_n$ converges as $n\to\infty$ as well.