inequality for the zeta function for real $s$

Question

I am wondering why $\zeta(s)<1+\frac{1}{2^s}+(s-1)^{-1} 2^{-s+1}$ for $s>1.$ I was able to show that $\zeta(s) \leq \frac{1}{1-\frac{1}{2^{s-1}}}$ but thats a much bigger upper bound. Does someone have a hint ?