I'm trying to prove the following inequality. Let $ s = x+iy \in \mathbb{C} $. Prove that
$$ \left\lvert\frac{1}{n^s} - \frac{1}{(n+1)^s}\right\rvert \leq \frac{\lvert s\rvert}{n^{x+1}} $$
The problem is I'm not very familiar with complex analysis. I've tried to write it down with the definition $ n^s = e^{s\log{n}} = e^{x\log{n}}(\cos(y\log{n}) + i\sin(y\log{n})) $, but it has lead me nowhere
The fundamental theorem of calculus tells you that
$$f(n+1) - f(n) = \int_n^{n+1} f'(t)\,dt$$
for continuously differentiable functions $f\colon (0,+\infty) \to \mathbb{C}$. You get the estimate
$$\lvert f(n+1) - f(n)\rvert \leqslant \sup \{ \lvert f'(t)\rvert : t \in [n,n+1]\}.$$
All that you then need is to find
$$\lvert t^z\rvert$$
for real positive $t$ and complex $z$.