Problem 6
Let \(\boldsymbol{K}\) be a metric compactum and \(\left\\{f_{n}(x)\right\\}\) a sequence of continuous functions on \(\mathrm{K}\), increasing in the sense that $$ f_{1}(x) \leqslant f_{2}(x) \leqslant \cdots \leqslant f_{n}(x) \leqslant \cdots $$ Prove that if \(\left\\{f_{n}(x)\right\\}\) converges to a continuous function on \(\mathrm{K}\), then the covergence is uniform (Dini's theorem).
Problem 9
Given a metric space \(\mathrm{R}\) with metric \(\rho\), let \(\boldsymbol{\Gamma}\) be a curve in \(R\) of finite length \(S\) with parametric representation $$ P=f(t) \quad(a \leqslant t \leqslant b) $$ Lets \(=\varphi(T)\) be the length of the arc $$ P=f(t) \quad(a \leqslant t \leqslant T) $$ (where \(\mathrm{T} \leqslant \mathrm{b}\) ), i.e., the arc of \(\Gamma\) going from the "initial point" \(P_{a}=\mathbf{f}(\mathrm{a})\) to the "final point" \(P_{T}=\mathrm{f}(\mathrm{T})\). Then \(\Gamma\) has a parametric representation of the form $$ P=g(s) \quad(0 \leqslant s \leqslant S) $$ where \(g(s) \equiv \mathbf{f}\left(\varphi^{-1}(s)\right)\) if \(\varphi\) is one-to-one. Prove that $$ \rho\left(g\left(s_{1}\right), g\left(s_{2}\right)\right) \leqslant\left|s_{1}-s_{2}\right| $$ Hint. The length of an arc is no less than the length of the inscribed chord.
Problem 11
Given a compact metric space \(\mathrm{R}\), suppose two points A and B in \(\mathrm{R}\) can be joined by a continuous curve of finite length. Prove that among all such curves, there is a curve of least length. Comment. Even in the case where \(R\) is a "smooth" (i.e., sufficiently differentiable) closed surface in Euclidean 3-space, this result is not amenable to the methods of elementary differential geometry, which ordinarily deals only with the case of "neighboring" points A and B.
