I'm analyzing the following isometric immersion of $\mathbb H^2$ in $\mathbb R^\infty$ given by $f(x,y)=(x_1,x_2,\dots,x_{2m-1},x_{2m},\dots)$ with 
\begin{align}\label{5.1}
    x_{2m-1}=\color{red}{2}\operatorname{Re}\frac{(x+iy)^m}{\sqrt{m}},\quad x_{2m}=\color{red}{2}\operatorname{Im}\frac{(x+iy)^m}{\sqrt{m}},\quad m=1,2,\dots
\end{align}
I tried to check that it really is an isometric immersion, but I cannot calculate $f^*g_{\mathbb R^\infty}=g$, some metric $g$, or give it shape, I have tried to do it by means of its polar representation but I have gotten confused without reaching anything concrete. Any ideas how to attack this problem?

[Here][1] I leave the original document.

  [1]: https://www.e-periodica.ch/cntmng?type=pdf&pid=com-001:1932:4::24