Proving the completion of a metric space is complete

In summary, Rudin's problem asks to find a Cauchy sequence of equivalence classes that converges to a specific one. Assuming the class is Cauchy, the sequence will converge. However, it is possible to substitute different sequences that are also in the same equivalence class, so the sequence will stay the same.
  • #1
tbrown122387
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Homework Statement



Having a little trouble on number 24 of Chapter 3 in Rudin's Principles of Mathematical Analysis. How do I prove that the completion of a metric space is complete?



Homework Equations



[itex]X[/itex] is the original metric space, [itex]X^* [/itex] is the completion, or the set of equivalence classes generated by the metric [itex]\bigtriangleup(P,Q) = \lim_{n \to \infty} d(p_n, q_n) [/itex] where [itex]P,Q \in X^* [/itex] and [itex] \{p_n\} \in P[/itex] and [itex]\{q_n\} \in Q. [/itex]


The Attempt at a Solution



I guess the thing that's confusing me is thinking about Cauchy sequences of equivalence classes. Every time you compare two new equivalence classes, you compare the limit of two real number sequences I guess. Am I thinking about this correctly?

My gut instinct is to use Baire's theorem for this. Maybe construct some shrinking neighborhoods and show the infinite intersection is nonempty? If this is a good path to take, I'll have two questions:
1. How to construct the neighborhoods (maybe let [itex] N_{r_n}p_n[/itex] be the smallest neighborhood containing the previous point, centered at [itex] p_n [/itex])?

2. How to show these neighborhoods are dense in [itex] X^* [/itex]? How can an equivalence class be a limit or a point of a neighborhood?
 
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  • #2
Tell me if you think this works:

Assume [itex]\{P_n\}[/itex] is Cauchy in [itex] X^* .[/itex] Fix [itex] \epsilon > 0 [/itex] and pick [itex] N [/itex] large enough so that [itex] n \ge N [/itex] implies [tex] \bigtriangleup (P_n, P_m) < \epsilon .[/tex] This means that [itex] \lim_{n \to \infty} d(p_n, p_m) < \epsilon [/itex] for [itex] \{p_n\}, \{p_m\}[/itex] in [itex] P_n, P_m [/itex] respectively. Since [itex] \epsilon [/itex] was arbitrary, [itex] \{p_n\}, \{p_m\} [/itex] are equivalent, meaning they're in the same equivalence class. Name this class [itex] P. [/itex] Clearly [tex] \bigtriangleup (P_n, P) = 0 [/tex] since for any [itex] \{p_j\} \in P, \lim d(p_n, p_j) = 0 [/itex] (by 24(b)). So, Cauchy sequences of equivalence classes converge to some specific equivalence class.

Note: in 24(b) we showed that it if you substitute in different sequences from the same equivalence class, [itex] \bigtriangleup [/itex] stays the same.
 

Related to Proving the completion of a metric space is complete

1. What is a metric space?

A metric space is a mathematical concept that defines a set of objects and a distance function between those objects. This distance function, also known as a metric, satisfies certain properties such as being non-negative, symmetric, and satisfying the triangle inequality.

2. What does it mean for a metric space to be complete?

A metric space is complete if every Cauchy sequence in the space converges to a point within the space. In simpler terms, this means that in a complete metric space, there are no "missing" points and all possible limits of sequences are contained within the space.

3. How do you prove the completion of a metric space?

The most common method for proving the completion of a metric space is by constructing a Cauchy completion. This involves adding points to the original metric space to fill in the "missing" points and ensuring that all Cauchy sequences in the original space converge to a point in the completion. The completeness of the completion can then be proven using the properties of Cauchy sequences.

4. Why is it important to prove the completion of a metric space?

Proving the completion of a metric space is important because it allows us to extend the properties and theorems of the original space to the completion. This can be useful in solving problems and proving theorems in mathematics and other fields that involve metric spaces.

5. Are all metric spaces complete?

No, not all metric spaces are complete. In fact, most metric spaces are not complete. For example, the set of rational numbers with the usual metric is not complete. However, it is possible to construct a completion for any metric space, making it complete.

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