IMC 2010 · Problem 2

Let $a_0, a_1, \dots, a_n$ be positive real numbers such that $a_{k+1} - a_k \ge 1$ for all $k = 0, 1, \dots, n-1$. Prove that

$$1 + \frac{1}{a_0}\left( 1 + \frac{1}{a_1 - a_0} \right) \cdots \left( 1 + \frac{1}{a_n - a_0} \right) \le \left( 1 + \frac{1}{a_0} \right)\left( 1 + \frac{1}{a_1} \right) \cdots \left( 1 + \frac{1}{a_n} \right).$$