Consider the function $$g: \mathbb{R}\to \mathbb{R}^\omega$$ given by $$g(t)=(t,t,t,...)$$ where $ \mathbb{R}^\omega$ is in the uniform topology. Can we find the exact answer to $$g^{-1}(B_{\rho}((1-\frac{1}{2^n})_{n\in \mathbb{N}}, 1))$$ where $\rho$ is the uniform metric as defined in Munkres' Topology book (page ~120)) i.e. $$\rho(x,y) = \sup_i\{ \min\{|x_i - y_i|, 1\}\}$$
I have been able to show that the pre-image is of the form $(0,1+\epsilon)$ but I can't determine the exact value of the right hand side limit.
Regards.