Rewrite the given and the required inequalities as $\frac{1}{x} + \frac{1}{y} + \frac{1}{z} \le 1$ and $\frac{1}{\sqrt{xy}} + \frac{1}{\sqrt{xz}} + \frac{1}{\sqrt{yz}} \le 1$.

By the inequality of means, we have
$$xy + xz \geq 2\sqrt{xy \cdot xz}, \quad xy + yz \geq 2\sqrt{xy \cdot yz}, \quad xz + yz \geq 2\sqrt{xz \cdot yz}.$$
Add these three inequalities and divide the result by $2$. Taking into account the condition, we get
$$xyz \geq xy + xz + yz \geq x\sqrt{yz} + y\sqrt{xz} + z\sqrt{xy}.$$
Dividing the obtained inequality by $\sqrt{xyz}$, we obtain the required result.