I think there is an issue in this proof, where $i(n)$ and $k(n)$ need to be redefined slightly as follows for the intersection of the nested open balls to be a singleton: $i(n) = \inf \{m > n : \overline{B(x_m,2^{-m})} \subset B(x_n,2^{-n}) \setminus B(x_n,2^{-m}) \}$ and $k(n)$ redefined similarly.
By making these changes reading up to the third image, the proof makes sense to me, but I am totally at a loss as to why in the third image (last paragraph) of this proof we have that $r_n \leq \rho(y_{\sigma},y_{\tilde{\sigma}})$.
Any help illuminating why this is the case would be much appreciated. This lower bound is used to prove the continuity of the inverse map and so I am assuming it must be true (or a very close lower bound is true).
Edit : Now that I think about this a little more, the cantor space is compact, and since $f: C \rightarrow f(C)$ is a continuous bijection from a compact space to $f(C)$ which is hausdorff, isn't $f$ automatically a homeomorphism? In which case the final paragraph proving the continuity of the inverse map seems unnecessary, but my question still stands as to how the lower bound mentioned in the final paragraph was achieved.
The proof quoted as an image can be found here https://www.math.ucla.edu/~biskup/245b.1.20w/ (lecture 15 - second to last page)
I think that you’re right about the definitions of $i(n)$ and $k(n)$; I don’t see anything in the construction that could substitute for that change. Fortunately, the extra requirement causes no difficulty. I think that there’s also a small error in $(15.38)$: if I’m not mistaken, it should read
$$r_{\color{red}{n+1}}\le\rho(y_\sigma,y_{\tilde{\sigma}})\le 2^{-n+1}\,:$$
if $\rho(y_\sigma,y_{\tilde{\sigma}})$ were less than $r_{n+1}$, $\sigma$ and $\tilde{\sigma}$ would agree in the first $n+1$ places, but $\sigma_{n+1}\ne\tilde{\sigma}_{n+1}$.
Finally, the last paragraph is indeed superfluous; it may be that the author has chosen not to assume that much topological knowledge on the part of the reader.