BALTIC-WAY 2000 Problem 20

For every positive integer $n$, let

$$x_{n}=\frac{(2 n+1) \cdot(2 n+3) \cdots \cdots \cdot(4 n-1) \cdot(4 n+1)}{2 n \cdot(2 n+2) \cdots \cdots \cdot(4 n-2) \cdot 4 n}$$

Prove that $\frac{1}{4 n}<x_{n}-\sqrt{2}<\frac{2}{n}$.