Given $$x_1=\arctan2>x_2>x_3\dots\\ \sin(x_n-x_{n-1})=\frac1{2^{n+1}}\sin x_n \sin{x_{n-1}} $$
Find $\cot x_n$ and prove $\lim_{x\to\infty}x_n=\frac\pi4$
I found that $\cot x_n=\cot x_{n-1}-(1/{2^{n+1}})$
Im not sure how to proceed with the limit. If I put $x_n=x_{n-1}$ in above eq it evaluate to $0=0$