Problem 1
Let \(M\) be a totally bounded subset of a metric space \(R\). Prove that the \varepsilon-nets figuring in the definition of total boundedness of \(\mathrm{M}\) can always be chosen to consist of points of \(\mathrm{M}\) rather than of \(\mathrm{R}\). Hint. Given an \(\varepsilon\)-net for M consisting of points \(a, a, \ldots, a, \in R\), all within \(\varepsilon\) of some point of \(M\), replace each point \(a\), by a point \(b\), EM such that \(\rho\left(a_{k}, b_{k}\right) \leqslant E\).
Problem 2
Prove that every totally bounded metric space is separable. Hint, Construct a finite \((1 / n)\)-net for every \(\mathrm{n}=1,2, \ldots\) Then take the union of these nets.
Problem 3
Let \(M\) be a bounded subset of the space \(C_{[a, b]}\). Prove that the set of all functions with \(\mathbf{f} \in \mathrm{M}\) compact. $$ F(x)=\int_{a}^{x} f(t) d t $$
