Let $\{x_n\}$ and $\{y_n\}$ be two real number sequences which are defined as follow: $$x_1=y_1=\sqrt{3},\quad x_{n+1}=x_n +\sqrt{1+x_n^2},\quad y_{n+1}=\frac{y_n}{1+\sqrt{1+y_n^2}}$$ for all $n\ge 1$. Prove that $2 < x_ny_n < 3$ for all $n>1$.