Alternating Riemann Zeta Function (Dirichlet eta function) convergence proof

Question

I am reading a proof on proving the convergence of the alternating Riemann Zeta Function but I cant understand how they went from the 2nd line of math to the 3rd. Can someone please explain why the inequality $\left| \frac{1}{(2n-1)^s}-\frac{1}{(2n)^s}\right|\leq\left|\frac{s}{(2n-1)^{s+1}}\right|$ is true? I understand everything else but why this inequality is true.

Answer 1

Triangle inequality:

$$\left|\int_{2n-1}^{2n} sx^{-s-1} \,\mathrm{d}x\right|\le \int_{2n-1}^{2n} |sx^{-s-1}|\,\mathrm{d}x.$$

Now bound the second integral by replacing the integrand with its maximum on the interval.