My notes define Q-linear convergence (for simplicity here for a positive sequence $(x_n)$ converging to $0$) as: $$\exists\sigma\in (0,1) \text{ s.t. }\frac{x_{k+1}}{x_k}\leq 1-\sigma, \forall k\in\mathbb{N}$$
I can't seem to find this definition anywhere online. Is the $\forall k$ meant to be before or after the $\exists\sigma$? i.e. which of the following two is the correct definition (and is this equivalent to the usual definition of Q-linear convergence?):
$$(1)\ \ \forall k\in\mathbb{N},\exists\sigma_k\in (0,1) \text{ s.t. }\frac{x_{k+1}}{x_k}\leq 1-\sigma_k$$ $$(2)\ \ \exists\sigma\in (0,1) \text{ s.t. } \forall k\in\mathbb{N}, \frac{x_{k+1}}{x_k}\leq 1-\sigma$$