The USSR Olympiad Problem Book
#145. Prove that, for any natural number \(n\), the following is true:
\begin{align*}
2 < \left(1+\frac{1}{n}\right)^n < 3.
\end{align*}
Hint. For the upper limit, expand and use:
\begin{align*}
\sum_{k=1}^\infty \frac{1}{k!} < 2.
\end{align*}