How can I prove that the sequence $\{x_k\}$ where $x_1 = 4, x_k = x_{k-1} + 4(\frac{1}{4k-3} - \frac{1}{4k-5})$ is Cauchy?
I understand the definition of Cauchy and I can see that the sequence is indeed Cauchy, but I think I am getting lost in messy algebra here to prove it. I start with setting up $|x_m - x_n|$ and I want to use the clever $\frac{\epsilon}{2}$ trick, but I am struggling to work through it. Any assistance would be greatly appreciated.
Show that $x_k$ is decreasing whereas $x_k-4\frac1{4k-3}$ is increasing.