I don't know where the comes from, but there is a fairly simple way to prove a stronger inequality.

Let . Integrate by parts to get . Therefore .

Using that recurrence relation, you can show by induction that . In fact, the base case n=0 is obvious. The inductive step amounts to the inequality , which you can check by squaring both sides.

Since is clearly less than , that is a better inequality than what was asked for.