In Multiplicative Number Theory, Montgomery and Vaughan provide an identity for Riemann's zeta function. $$\zeta(s)=\frac{s}{s-1}-s\int_{1}^{\infty}\left \{ u \right \}u^{-s-1}du$$ They then state
But it seems to me there are several steps missing. Why does that sufficies to prove the inequality? Why are they only using the $\int_{1}^{\infty}\left \{ u \right \}u^{-s-1}du$ part of the identity? How does that prove the original inequality?
The missing step is to express this intergral with the fractional part in terms of the zeta function, using original identity: $$\int_1^\infty\{u\} u^{-\sigma- 1}du = \frac{1}{\sigma - 1} - \frac{\zeta(\sigma)}{\sigma}.$$
Then plug-in into the chain of inequalities: $$0 \le \frac{1}{\sigma - 1} - \frac{\zeta(\sigma)}{\sigma} < \frac{1}{\sigma}.$$ By re-arranging we obtain what's required. A minor point is that we requiry the first inequality in the chain to be strict, but clearly this is true.