The USSR Olympiad Problem Book
#149.* Prove that if \(m\) > \(n\) (where \(m, \; n\) are natural numbers):
\begin{align*}
(a) \quad \hspace{20pt} \left(1+\frac{1}{m}\right)^m & > \left(1+\frac{1}{n}\right)^n.\\[0.7em]
(b) \hspace{20pt} \left(1+\frac{1}{m}\right)^{m+1} & < \left(1+\frac{1}{n}\right)^{n+1}, \quad (n \geq 2).
\end{align*}
Hint. Proving the result for \(m = n+1\) is sufficient. Use:
\begin{align*}
(n+1)a^n < \frac{b^{n+1} - a^{n+1}}{b-a} < (n+1)b^n, \quad 0 \leq a < b, \; n = 1, 2, 3, \ldots
\end{align*}