is $\zeta(s) = \frac{1}{1-2^{1-s}}\eta(s)$ an analytic continuation of $\zeta(s)$ for $\sigma > 0$

Question

It seems from what I have read on the net, that the above representation of $\zeta(s)$ is a valid analytic continuation of $\sum_{i=1}^{\infty}\frac{1}{i^s}$ for $\sigma > 0$ except for a simple pole at $s=1$.

Is this not true?

Thanks