Proof for R2757: Proper contraction map is continuous
Let $X$ and $Y$ each form a D2506: Topological metric space.
Let $d_X$ and $d_Y$ each be the D58: Metric in $X$ and $Y$, respectively.
Let $f : X \to Y$ be a D49: Proper contraction with respect to $X$ and $Y$.
This result is a corollary to the results
(i) R2756: Proper contraction map is Lipschitz
(ii) R80: Lipschitz map is continuous