# Metric Spaces: Complete Metric Spaces

Cauchy criterion: for all $$\epsilon > 0$$ there exists $$N_\epsilon$$ such that $$\rho(x_n, x_{n^\prime}) < \epsilon$$ for all $$n, n^\prime > N_\epsilon$$. A sequence satisfying this criterion is called a Cauchy sequence or fundamental.

Theorem 1

Every convergent sequence $$\{x_n\}$$ is fundamental.

Complete if every Cauchy sequence in $$R$$ converges to an element of $$R$$. Incomplete otherwise.

Theorem 2 (Nested sphere theorem) Complete if and only if every nested sequence $\{S_n\} = \{S_\infty [x_n, r_n]\}$ of closed spheres such that $$r_n \rightarrow 0$$ as $$n \rightarrow \infty$$ has a nonempty intersection $$\cap_{n=1}^\infty S_n$$.

Theorem 3 (Baire) A complete metric space cannot be represented as the union of a countable number of nowhere dense sets.

$$R^*$$ completion of $$R$$ if $$R \subseteq R^*$$ and $$[R] = R^*$$.

Theorem 4 Every metric space $$R$$ has a completion. This completion is unique to within an isometric mapping carrying every point $$x \in R$$ into itself.

## Problems

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 $$t_0$$.

Do as the hint says.

2. Prove that the space $$m$$ is complete.

$$\rho(x_n, x_{n^\prime}) = \sup_k (x_{n_k} - x_{{n^\prime}_k}) < \epsilon$$

3. Prove that if $$R$$ is ocmplete, then the intersection $$\cap_{n = 1}^\infty S_n$$ riguring in Theorem 2 consists of a single point.

$$R$$ complete, so limit in $$R$$ by definition. And by Theorem 2, intersection nonempty.

4. The diameter of a subset $$A$$ of a metric space $$R$$ is $d(A) = \sup_{x, y \in A} \rho(x, y)$ Suppose $$R$$ complete, and let $$\{A_n\}$$ be a sequence of nested of closed subsets of $$R$$. Suppose further that $$\lim_{d \rightarrow \infty} d(A_n) = 0$$. Prove that the intersection $$\cap_{n = 1}^\infty A_n$$ is nonempty.

Same as the nested sphere theorem. Also use definition of complete.

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.

There’ll always still be a lowest and hightest number.

6. Give an example of a complete metric space $$R$$ and a nested sequence $$\{A_n\}$$ of closed subsets of $$R$$ such that $$\cap_{n=1}^\infty A_n = \emptyset$$. Reconcile this example with Problem 4.

Take all of the subsets to be the empty set. The diameter limit no longer holds.

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

Both ways by definitions.

8. Prove that the real line equipped with the distance $\rho(x, y) = |\arctan x - \arctan y|$ is an incomplete metric space.

$$\frac{\pi}{2}$$.

9. Give anxample of a complete metric space homeomorphic to an incomplete metric space.

$$f(x) = \frac{2}{\pi} \arctan x$$

10. Construct the real number system.

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 $$\{x_n + y_n\}$$.

Also need to define the products, and verify that the usual axioms of arithmetic are satisfied.