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*}