Proof Writing & Problem Solving · Challenge problem

Let $P_n = \dfrac{1}{2} \cdot \dfrac{3}{4} \cdot \dfrac{5}{6} \cdots \dfrac{2n - 1}{2n}$. The bound $P_n \le \dfrac{1}{\sqrt{3n}}$ is true, but its inductive step cannot be completed; the STRONGER statement $P_n \le \dfrac{1}{\sqrt{3n + 1}}$ can be proved by induction. What upper bound does the stronger statement give for $P_{16}$?