Problem 1
Prove that the limit \(\mathbf{f}(t)\) of a uniformly convergent sequence of functions \(\left\\{f_{n}(t)\right\\}\) continuous on \([a, b]\) is itself a function continuous on \([a, b] .\) Hint. Clearly $$ \left|f(t)-f\left(t_{0}\right)\right| \leqslant\left|f(t)-f_{n}(t)\right|+\left|f_{n}(t)-f_{n}\left(t_{0}\right)\right|+\left|f_{n}\left(t_{0}\right)-f\left(t_{0}\right)\right| $$ where \(t, t_{0} \in[\mathrm{a}, \mathrm{b}]\). Use the uniform convergence to make the sum of the first and third terms on the right small for sufficiently large \(\mathrm{n}\). Then use the continuity of \(f_{n}(t)\) to make the second term small for \(t\) sufficiently close to \(t_{0}\).
Problem 4
By the diameter of a subset \(A\) of a metric space \(R\) is meant the number $$ d(A)=\sup _{x, y \in A} p(x, y) $$ Suppose \(R\) is complete, and let \(\\{A,\), be a sequence of closed subsets of \(R\) nested in the sense that $$ A_{1} \supset A_{2} \supset \ldots \supset A_{n} \supset \cdots $$ Suppose further that $$ \lim _{n \rightarrow \infty} d\left(A_{n}\right)=0 $$ Prove that the intersection \(\bigcap_{n=1}^{\sim} A\), is nonempty.
Problem 5
\(A\) subset \(A\) of a metric space \(R\) is said to be bounded if its diameter \(d(A)\) is finite. Prove that the union of a finite number of bounded sets is bounded.
Problem 7
Prove that a subspace of a complete metric space \(\mathrm{R}\) is complete if and only if it is closed.
Problem 8
Prove that the real line equipped with the distance $$ \rho(x, y)=|\arctan x-\arctan y| $$ is an incomplete metric space.
