# 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

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.

Prove that the space \(m\) is complete.

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

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.

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.

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.

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.

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

Both ways by definitions.

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

\(\frac{\pi}{2}\).

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

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

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.