Let $X\subseteq \mathbb{R}^{2}$ be the subset $\bigcup_{n\in\mathbb{Z}}X_n$ where $X_n=\{n\}\times [0, 2^{-|n|}]$, with the metric inherited from $\mathbb{R}^{2}$, and define $f:X\rightarrow X$ by
$ f(n,y):= \left\lbrace \begin{array}{c l} {{(n+1,2y)}} & \text{if $n<0$},\\ {(n+1,\frac{y}{2})} & \text{if $n\geq0$}. \end{array} \right. $
Also $Y=\cup _{n\in\mathbb{Z}}\{n\}\times [0, 1]$ and $g$ be a translation on the first coordinate and identity on the second.
In a paper, author said that there is a homeomorphism $h:X\to Y$ such that $h\circ f= g\circ h$ (without proof). I don't understand it. Please help me