Lemma. For $x,y \geq 1$, $\left( x- \frac{1}{x} \right) + \left( y - \frac{1}{y} \right) \leq \left( xy- \frac{1}{xy} \right)$.
Proof. $\iff x^2 y + xy^2 - x - y \leq x^2 y^2 - 1 \iff (y^2 -1)\left( x - \frac{1}{2} \right)^2 + (x^2 -1) \left( y - \frac{1}{2} \right)^2 \geq 0$. $\blacksquare$
Note that $a_2, \cdots, a_n >1$ and $a_1 = (a_2 \cdots a_n)^{-1}$. The given inequaility is equivalent to
\[\sum_{i=2}^n \left( a_i - \frac{1}{a_i} \right) \leq \left( a_2 \cdots a_n - \frac{1}{a_2 \cdots a_n} \right)\]It can be proved by inductively applying the Lemma.