Let $\zeta$ be the Riemann zeta function, I am trying to show that it does not have a zero on the interval $[0, 1]$. I have tried using the extension of the $\zeta$ function to $Re(s)>0$ namely: $$ \zeta(s) = \frac{s}{s-1} - s\int_1^\infty \{x\}x^{-s-1}dx$$ However I could only bound this below by some negative constant which is clearly not helpful here...
Any hints would be much appreciated.
Hint: use the Dirichlet eta function $$\eta (s) = \sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n^s}=1-\frac{1}{2^s}+\frac{1}{3^s}-\dots$$ which is valid for $\operatorname{Re} s>0$, and satisfies the relation $$\eta(s)=({1-2^{1-s}})\zeta(s) $$