# Identity involving the zeta function

This might be very trivial, but a proof I'm reading on the bounds of the zeta-function uses the following fact:

If $s=\sigma+it$ is a complex number and if $\sigma\geq 2$, then $|\zeta(s)|\geq 1-\displaystyle\sum_{n\geq 2}\frac{1}{n^2}$.

What would be a rigorous explanation of this? Many thanks in advance.

It's just the triangle inequality together with $\lvert n^{s}\rvert = n^{\operatorname{Re} s}$ and the monotonicity of $\sigma \mapsto n^\sigma$. Thus we have