Question

Let \(X\) be a metric space. (a) Call two Cauchy sequences \(\left\\{p_{n}\right\\},\left\\{q_{n}\right\\}\) in \(X\) equivalent if $$\lim _{n \rightarrow \infty} d\left(p_{n}, q_{n}\right)=0$$ Prove that this is an equivalence relation. (b) Let \(X^{*}\) be the set of all equivalence classes so obtained. If \(P \in X^{*}, Q \in X^{*}\), \(\left\\{p_{n}\right\\} \in P,\left\\{q_{n}\right\\} \in Q\), define $$\Delta(P, Q)=\lim _{n \rightarrow \infty} d\left(p_{n}, q_{n}\right)$$ by Exercise 23, this limit exists. Show that the number \(\Delta(P, Q)\) is unchanged if \(\left\\{p_{n}\right\\}\) and \(\left\\{q_{*}\right\\}\) are replaced by equivalent sequences, and hence that \(\Delta\) is a distance function in \(X^{*}\). (c) Prove that the resulting metric space \(X^{*}\) is complete. (d) For each \(p \in X\), there is a Cauchy sequence all of whose terms are \(p ;\) let \(P\), be the element of \(X^{*}\) which contains this sequence. Prove that $$\Delta\left(P_{\nu}, P_{4}\right)=d(p, q)$$ for all \(p, q \in X .\) In other words, the mapping \(\varphi\) defined by \(\varphi(p)=P\), is an isometry (i.e., a distance-preserving mapping) of \(X\) into \(X^{\bullet}\). (e) Prove that \(\varphi(X)\) is dense in \(X^{*}\), and that \(\varphi(X)=X^{*}\) if \(X\) is complete. By \((d)\), we may identify \(X\) and \(\varphi(X)\) and thus regard \(X\) as embedded in the complete metric space \(X^{*}\). We call \(X^{*}\) the completion of \(X\).

Short answer

In conclusion, we have proved the following properties for the given metric space \(X\): (a) The equivalence relation defined between two Cauchy sequences is indeed an equivalence relation, as it satisfies reflexivity, symmetry, and transitivity. (b) The defined function \(\Delta(P,Q)\) is unchanged when equivalent sequences are considered, and it functions as a distance function in the space \(X^{*}\). (c) The metric space \(X^{*}\) is complete, as every Cauchy sequence in \(X^{*}\) converges to a point in \(X^{*}\). (d) The mapping \(\varphi\) is an isometry for all \(p,q \in X\), meaning that it preserves distances between points in the space. (e) The set \(\varphi(X)\) is dense in \(X^{*}\). Furthermore, if \(X\) is complete, then \(\varphi(X) = X^{*}\). These results provide a comprehensive understanding of the relationships between sequences and equivalence classes in the metric space \(X\). They are essential for studying the convergence and continuity properties of functions in metric spaces and their completion.

Step-by-step solution

(a) Proving the equivalence relation

To show that the given relation is an equivalence relation, we need to prove that it satisfies reflexivity, symmetry, and transitivity. 1. Reflexivity: For any Cauchy sequence \(\left\{p_n\right\}\), the distance between its elements is \(d(p_n, p_n)=0\). As \(\lim_{n \to \infty} 0 = 0\), the relation is reflexive. 2. Symmetry: If \(\lim _{n \rightarrow \infty} d\left(p_{n}, q_{n}\right)=0\) for two Cauchy sequences \(\left\{p_n\right\}\) and \(\left\{q_n\right\}\), then by the properties of distance functions, \(d\left(p_{n}, q_{n}\right) = d\left(q_{n}, p_{n}\right)\). Hence, \(\lim _{n \rightarrow \infty} d\left(q_{n}, p_{n}\right)=0\). The relation is symmetric. 3. Transitivity: If \(\lim _{n \rightarrow \infty} d\left(p_{n}, q_{n}\right)=0\) and \(\lim _{n \rightarrow \infty} d\left(q_{n}, r_{n}\right)=0\), then, applying the triangle inequality, we have: \(d(p_n, r_n) \le d(p_n, q_n) + d(q_n, r_n)\). As \(n \to \infty\), both \(d(p_n, q_n)\) and \(d(q_n, r_n)\) tend to 0, implying \(d(p_n, r_n) \to 0\). Thus, the relation is transitive. Since the relation is reflexive, symmetric, and transitive, it is an equivalence relation.

(b) Proving \(\Delta(P,Q)\) is a distance function in \(X^{*}\)

Suppose \(\left\{p_n'\right\}\) and \(\left\{q_n'\right\}\) are equivalent to \(\left\{p_n\right\}\) and \(\left\{q_n\right\}\) respectively. Then, \(\lim_{n\to\infty}d(p_n,p_n')=0\) and \(\lim_{n\to\infty}d(q_n,q_n')=0\). Now we can apply the triangle inequality: \(d(p_n',q_n')\le d(p_n',p_n)+d(p_n,q_n)+d(q_n,q_n')\). Taking the limit as \(n\to\infty\), we have: \(\lim_{n\to\infty}d(p_n',q_n')\le \lim_{n\to\infty}\left(d(p_n',p_n)+d(p_n,q_n)+d(q_n,q_n')\right)\). As both \(\lim_{n\to\infty} d(p_n,p_n')=0\) and \(\lim_{n\to\infty} d(q_n,q_n')=0\), the right-hand side converges to \(\lim_{n \rightarrow \infty} d(p_n, q_n) = \Delta(P,Q)\). Thus, we find that \(\lim_{n\to\infty}d(p_n',q_n')=\Delta(P,Q)\) as well. This implies that the value of \(\Delta(P,Q)\) is unchanged when \(\left\{p_n\right\}\) and \(\left\{q_n\right\}\) are replaced by equivalent sequences. Additionally, since \(\Delta(P,Q)=\lim _{n \rightarrow \infty} d\left(p_{n}, q_{n}\right)\), it satisfies the properties of distance functions such as non-negativity, identity of indiscernibles, and the triangle inequality. Hence, it is a distance function in \(X^{*}\).

(c) Proving \(X^*\) is complete

To prove that \(X^*\) is complete, we need to show that every Cauchy sequence in \(X^*\) converges to a point in \(X^*\). Let \(\left\{P_n\right\}\) be a Cauchy sequence in \(X^*\), and \(\left\{p_n^k\right\}\in P_k\) be Cauchy sequences with elements in \(X\) belonging to each \(P_k\). Since \(\left\{P_n\right\}\) is Cauchy in \(X^*\), for any given \(\varepsilon>0\), we can find an \(N_0\) such that for all \(n,m\ge N_0\), \(\Delta(P_n,P_m)<\varepsilon\). Now, consider sequences \(\left\{p_n^{N_0}\right\}\) and \(\left\{p_m^{N_0}\right\}\). We have: $$\lim_{n,m\to\infty}\Delta(P_n,P_m)=\lim_{n,m\to\infty}(\lim_{k\to\infty}d(p_n^k,p_m^k))=\lim_{k\to\infty}(\lim_{n,m\to\infty}d(p_n^k,p_m^k)).$$ From the expression above, since \(\lim_{n,m\to\infty}\Delta(P_n,P_m)=\Delta(P_{N_0},P_{N_0})=0\), we have: $$\lim_{k\to\infty}\left(\lim_{n,m\to\infty}d(p_n^k,p_m^k)\right)=0.$$ Thus, the sequence \(\left\{p_{N_0}^k\right\}\) is also Cauchy since the limit of the sequence of distances converges to 0. Define a new equivalence class \(P_\infty\) containing \(\left\{p_{N_0}^k\right\}\). We claim that \(\lim_{n\to\infty}P_n=P_\infty\). For any \(\varepsilon>0\), we can find \(N\ge N_0\) such that for all \(n\ge N\), \(\Delta(P_n,P_{N_0})<\varepsilon\), which implies \(\Delta(P_n,P_\infty)<\varepsilon\). Therefore, every Cauchy sequence in \(X^*\) converges to an element in \(X^*\), which proves that \(X^*\) is complete.

(d) Proving \(\varphi(p)=P_p\) is an isometry

Let \(P_p\) and \(P_q\) be the elements in \(X^{*}\) that contain the Cauchy sequences \(\left\{p\right\}\) and \(\left\{q\right\}\) respectively. We have to prove that \(\Delta(P_p, P_q) = d(p, q)\) for all \(p, q \in X\). We already know that: $$\Delta(P_p,P_q)=\lim_{n\to\infty}d(p,q).$$ Since the sequences \(\left\{p\right\}\) and \(\left\{q\right\}\) are constant, their distance \(d(p,q)\) does not change as \(n\to\infty\). Therefore, \(\Delta(P_p,P_q)=d(p,q)\), and thus, \(\varphi(p)=P_p\) is an isometry.

(e) Proving \(\varphi(X)\) is dense in \(X^*\) and \(\varphi(X)=X^*\) if \(X\) is complete

To prove that \(\varphi(X)\) is dense in \(X^*\), we need to show that for every element \(P\in X^*\), and for any \(\varepsilon>0\), there exists \(p\in X\) such that \(\Delta(\varphi(p),P)<\varepsilon\). Let \(P\in X^*\) and let \(\left\{p_n\right\}\in P\). Choose \(p\in X\) such that \(d(p,p_n)<\varepsilon\) for some \(n\). Then, we have: $$\Delta(\varphi(p),P)=\Delta(P_p,P)\le\Delta(P_p,P_{p_n})+\Delta(P_{p_n},P),$$ where \(P_{p_n}\) is the element of \(X^*\) containing the sequence \(\left\{p_n\right\}\). Since \(\left\{p_n\right\}\in P\) and \(\Delta(P_p,P_{p_n})=d(p,p_n)\), we get: $$\Delta(\varphi(p),P)\le d(p,p_n)+\Delta(P_{p_n},P)=d(p,p_n).$$ As \(d(p,p_n)<\varepsilon\), this implies \(\Delta(\varphi(p),P)<\varepsilon\), showing that \(\varphi(X)\) is dense in \(X^*\). If \(X\) is complete, then every Cauchy sequence in \(X\) converges to a point in \(X\). Given \(P\in X^*\), there exists a Cauchy sequence \(\left\{p_n\right\}\), which converges to \(p\in X\). Then, we find that \(\Delta(P, \varphi(p)) = 0\), which implies \(P=\varphi(p)\), and thus \(\varphi(X)=X^*\).

Key concepts

Cauchy Sequences

Cauchy sequences are essential concepts in the study of metric spaces. They form a bridge between simple sequences and the notion of convergence without the need for specifying a limit. A sequence \( \{ p_n \} \) in a metric space is called Cauchy if, for every positive number \( \varepsilon \), there is an integer \( N \) such that for all \( m, n \geq N \), the distance \( d(p_m, p_n) < \varepsilon \). This definition essentially means that the terms of the sequence become arbitrarily close to each other as the sequence progresses.
Understanding Cauchy sequences opens the door to grasping the concept of completeness in metric spaces—where every Cauchy sequence converges to a limit within the space. This property is crucial because it ensures that mathematical operations and transformations performed within the space adhere to predictable outcomes.
Here are some key points about Cauchy sequences:

• They do not require the sequence to converge within the initial space; only that the terms get closer together.
• They are used to define completeness in metric spaces, which is vital for mathematical analysis.

Equivalence Relation

An equivalence relation on a set is a mathematical means to group elements into classes where each group, or equivalence class, shares a common property. In the context of Cauchy sequences, two sequences \( \{ p_n \} \) and \( \{ q_n \} \) are said to be equivalent if \( \lim_{n \to \infty} d(p_n, q_n) = 0 \).
For a relation to be an equivalence relation, it must satisfy three properties: reflexivity, symmetry, and transitivity. In other words, each sequence is equivalent to itself (reflexivity), if one sequence is equivalent to another, then the second is equivalent to the first (symmetry), and if one sequence is equivalent to another, which in turn is equivalent to a third sequence, then the first is equivalent to the third (transitivity).
This concept is important because it allows us to classify sequences in such a way that all sequences in the same class converge to the same limit or behave in a similar "limit-like" manner in terms of their distances.

• Equivalence relations help in structuring set elements into well-defined groups.
• They are foundational in creating quotient spaces, like \( X^* \), derived from metric spaces.

Distance Function

A distance function, or metric, gives a formal way to describe the distance between elements in a metric space. In this problem, for any two equivalence classes \( P, Q \) in \( X^* \), the distance function \( \Delta(P, Q) \) is defined by the limit \( \lim_{n \to \infty} d(p_n, q_n) \), where \( \{ p_n \} \) is a representative sequence from \( P \) and \( \{ q_n \} \) is from \( Q \).
The distance function must satisfy:

• Non-negativity: \( \Delta(P, Q) \geq 0 \) for any \( P, Q \).
• Identity of Indiscernibles: \( \Delta(P, Q) = 0 \) if and only if \( P = Q \).
• Symmetry: \( \Delta(P, Q) = \Delta(Q, P) \).
• Triangle Inequality: \( \Delta(P, Q) \leq \Delta(P, R) + \Delta(R, Q) \) for any \( P, Q, R \).

Properly defining the distance function ensures that \( X^* \) is a metric space, where we can logically talk about the closeness of elements, just like in the original space \( X \).

Completeness

Completeness of a metric space is the property that all Cauchy sequences converge within the space. This is a pivotal concept in mathematical analysis, which leads to significant results in spaces such as \( \mathbb{R} \), the set of real numbers, and extends to \( X^* \), the completion of metric space \( X \).
In the exercise, proving \( X^* \) is complete involves demonstrating that every Cauchy sequence of classes \( \{ P_n \} \) in \( X^* \) converges to an equivalence class within \( X^* \). Since any sequence in \( X^* \) is formed from limits of within \( X \), \( X^* \) consists of points defined as these equivalence classes.
Completeness is crucial for ensuring that limits and infinite processes result in outcomes that remain in the same mathematical context.

• Ensures functions and limits are well-defined.
• Central for analyzing convergence and stability in mathematical spaces.

Realizing the completeness of \( X^* \) allows us to use this "complete" space as a broader framework, embedding \( X \) within a more structured environment that has all requisite limits and processes accounted for.