equivalent metrics on shift spaces

Let $$\mathcal{M}=\prod_{\mathbb{Z}}\{1,\ldots,k\}$$, with finite $$k$$. On this space we define the following family of metrics:

Let $$a>1$$ and define $$\rho_a:\mathcal{M}\times \mathcal{M}\rightarrow [0,\infty)$$ as $$\rho_a(x,y)=\left\{ \begin{array}{ll} a^{-N} & \text{if x\neq y and N is maximal and } x_{[-N,N]}=y_{[-N,N]}\\ 0 & x=y \end{array}\right.$$ where $$x_{[-N,N]}$$ indicate the subword $$x_{-N}...x_{N}$$ of the word $$x=(x_i)_{i\in\mathbb{Z}}\in\mathcal{M}$$.

The work I am reading claims that the metrics $$\rho_a$$ are all equivalent. However, I am sceptical about it: let $$a>b>1$$, then clearly $$\rho_a(x,y)\leq \rho_b(x,y)$$ for all $$x,y\in\mathcal{M}$$, but how do we see that there exists a $$c>0$$ independent of $$x,y$$ such that $$\rho_a(x,y)\geq c \rho_b(x,y)$$? The quotient $$\rho_b(x,y)/\rho_a(x,y)$$ can be arbitrarily large, for example if $$b=a^\gamma$$ for $$0<\gamma<1$$.

What am I missing here?

Thanks a lot for the help!

For two metrics to be equivalent, it's not necessary that each is bounded by a constant times the other (this is a stronger condition) -- they simply have to generate the same topology. In this case you can verify that any ball $$B_r(x) = \{y \in \mathcal{M} : \rho_a(x, y) < r\}$$ under one $$\rho_a$$ is still a ball under any other $$\rho_b$$, so all $$\rho_a$$ are in fact equivalent.