Problem Set based on VII. Complete Metric Spaces

Problem 1.

Prove that the limit $f(t)$ of a uniformly convergent sequence of functions $\{ f_{n}(t)\}$ continuous on $[a,b]$ is itself a function continuous on $[a,b]$ .

Hint: Clearly,

$|f(t)-f(t_{0})| \leq |f(t) - f_{n}(t)|+|f_{n}(t) - f_{n}(t_{0})|+|f_{n}(t_{0}) - f(t_{0})|$ where $t, t_{0} \in [a,b]$ .

Use the uniform convergence to make the sum of the first and third terms on the right small for sufficiently large n. Then, use the continuity of $f_{n}(t)$ to make the second term small for t sufficiently close to $t_{0}$ .

Problem 2.

Prove that if R is complete, then the intersection $\bigcap_{n=1}^{\infty}S_{n}$ figuring in the theorem 2 consists of a single point. Theorem 2 is recalled here: Nested sphere theorem: A metric space R is complete if and only if nested sequence $S_{n} = \{ S[x_{n}, r_{n}]\}$ of closed spheres in R such that $t_{n} \rightarrow 0$ as $n \rightarrow \infty$ has a non empty intersection $\bigcap_{n=1}^{\infty}S_{n}$ .

Problem 3.

Prove that the space m is complete. Recall: Consider the set of all bounded infinite sequences of real numbers $x = (x_{1}, x_{2}, \ldots, x_{k}, \ldots)$ and let $\rho(x,y) = \sup_{k}|x_{k}-y_{k}|$ . This is a metric space which we denote by m.

Problem 4:

By the diameter of a subset A of a metric space R is meant the number $d(A) = \sup_{x, y \in A} \rho(x,y)$

Suppose R is complete and let $\{ A_{n}\}$ be a sequence of closed sets of R nested in the sense that

$A_{1} \supset A_{2} \supset \ldots \supset A_{n} \supset \ldots$

Suppose further that

$\lim_{n \rightarrow \infty} d(A_{n})=0$ .

Prove that the intersection $\bigcap_{n=1}^{\infty}A_{n}$ is non empty.

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 6.

Give an example of a complete metric space R and a nested sequence $\{ A_{n}\}$ of closed subsets of R such that

$\bigcap_{n=1}^{\infty}A_{n} = \phi$ .

Reconcile this example with Problem 4 here.

Problem 7.

Prove that a subspace of a complete metric space R is complete if and only if it is closed.

Problem 8.

Prove that the real line equipped with the distance metric function

$\rho(x,y) = |\arctan{x} - \arctan{y}|$ is an incomplete metric space.

Problem 9.

Give an example of a complete metric space homeomorphic to an incomplete metric space.

Hint: Consider the following example we encountered earlier: The function $y = f(x) = \frac{2}{\pi}\arctan{x}$ establishes a homeomorphism between the whole real line $(-\infty, \infty)$ and the open interval $(-1,1)$ .

Comment: Thus, homeomorphic metric spaces can have different metric properties.

Problem 10:

Carry out the program discussed in the last sentence of the following example:

Ir R is the space of all rational numbers, then $R^{*}$ is the space of all real numbers, both equipped with the distance $\rho(x,y) = |x-y|$ . In this way, we can “construct the real number system.” However, there still remains the problem of suitably defining sums and products of real numbers and verifying that the usual axioms of arithmetic are satisfied.

Hint: If $\{ x_{n}\}$ and $\{ y_{n}\}$ are Cauchy sequences of rational numbers serving as “representatives” of real numbers $x^{*}$ and $y^{*}$ respectively, define $x^{*} = y^{*}$ as the real number with representative $\{ x_{n} + y_{n}\}$ .