Let \((X, d)\) be a metric space. Show that there is an isometric imbedding \(h\) of \(X\) into a complete metric space \((Y, D)\), as follows. Let \(\tilde{X}\) denote the set of all Cauchy sequences
\[ \mathbf{x} = (x_1, x_2, \dots) \]
of points of \(X\). Define \(\mathbf{x} \sim \mathbf{y}\) if
\[ d(x_n, y_n) \to 0. \]
Let \([\mathbf{x}]\) denote the equivalence class of \(\mathbf{x}\); and let \(Y\) denote the set of these equivalence classes. Define a metric \(D\) on \(Y\) by the equation
\[ D([\mathbf{x}], [\mathbf{y}]) = \lim_{n \to \infty} d(x_n, y_n). \]
Show that \(\sim\) is an equivalence relation, and show that \(D\) is a well-defined metric.
Define \(h: X \to Y\) by letting \(h(x)\) be the equivalence class of the constant sequence \((x, x, \ldots)\):
\[ h(x) = [(x, x, \dots)]. \]
Show that \(h\) is an isometric imbedding.
Show that \(h(X)\) is dense in \(Y\); indeed, given \(\mathbf{x} = (x_1, x_2, \ldots) \in \tilde{X}\), show the sequence \(h(x_n)\) of points of \(Y\) converges to the point \([\mathbf{x}]\) of \(Y\).
Show that if \(A\) is a dense subset of a metric space \((Z, \rho)\), and if every Cauchy sequence in \(A\) converges in \(Z\), then \(Z\) is complete.
Show that \((Y, D)\) is complete.
Solution
1. Part (a): Equivalence relation and well-defined metric \(D\).Proof:
1.1. Equivalence relation:- Reflexivity: \(d(x_n, x_n) = 0 \to 0\).- Symmetry: \(d(x_n, y_n) = d(y_n, x_n) \to 0\).- Transitivity: \(d(x_n, z_n) \le d(x_n, y_n) + d(y_n, z_n) \to 0 + 0 = 0\).
1.2. Limit existence in \(\mathbb{R}\): For Cauchy sequences \(\mathbf{x}, \mathbf{y}\), the triangle inequality gives \(|d(x_n, y_n) - d(x_m, y_m)| \le d(x_n, x_m) + d(y_n, y_m) \to 0\). Thus \((d(x_n, y_n))_{n=1}^\infty\) is a Cauchy sequence in \(\mathbb{R}\), hence converges to a unique real limit.
1.3. Independence of representatives: If \(\mathbf{x} \sim \mathbf{x}'\) and \(\mathbf{y} \sim \mathbf{y}'\), then \(|d(x_n, y_n) - d(x'_n, y'_n)| \le d(x_n, x'_n) + d(y_n, y'_n) \to 0\), so \(\lim d(x_n, y_n) = \lim d(x'_n, y'_n)\).
1.4. Metric axioms:- \(D([\mathbf{x}], [\mathbf{y}]) \ge 0\) and \(D([\mathbf{x}], [\mathbf{y}]) = 0 \iff \lim d(x_n, y_n) = 0 \iff \mathbf{x} \sim \mathbf{y} \iff [\mathbf{x}] = [\mathbf{y}]\).- Symmetry: \(D([\mathbf{x}], [\mathbf{y}]) = D([\mathbf{y}], [\mathbf{x}])\).- Triangle inequality: \(D([\mathbf{x}], [\mathbf{z}]) = \lim d(x_n, z_n) \le \lim (d(x_n, y_n) + d(y_n, z_n)) = D([\mathbf{x}], [\mathbf{y}]) + D([\mathbf{y}], [\mathbf{z}])\).
2. Part (b): Isometric embedding \(h: X \to Y\).Proof:
2.1. For \(x, y \in X\), \(D(h(x), h(y)) = \lim_{n \to \infty} d(x, y) = d(x, y)\).
2.2. Thus \(h\) preserves all pairwise distances, which proves \(h\) is an isometric embedding (injective and topological embedding).
3. Part (c): Density of \(h(X)\) in \(Y\).Proof:
3.1. Let \([\mathbf{x}] \in Y\) where \(\mathbf{x} = (x_1, x_2, \dots) \in \tilde{X}\).
3.2. For each fixed \(k \in \mathbb{Z}_+\), \(D(h(x_k), [\mathbf{x}]) = \lim_{n \to \infty} d(x_k, x_n)\).
3.3. Given \(\varepsilon > 0\), choose \(N\) such that \(d(x_k, x_n) < \varepsilon\) for all \(k, n \ge N\).
3.4. Taking \(n \to \infty\) yields \(D(h(x_k), [\mathbf{x}]) \le \varepsilon\) for all \(k \ge N\).
3.5. Hence \(\lim_{k \to \infty} h(x_k) = [\mathbf{x}]\) in \((Y, D)\), proving \(h(X)\) is dense in \(Y\).
4. Part (d): Density criterion for completeness.Proof:
4.1. Let \((z_k)\) be a Cauchy sequence in \((Z, \rho)\).
4.2. Since \(A\) is dense, for each \(k\) choose \(a_k \in A\) with \(\rho(a_k, z_k) < \frac{1}{k}\).
4.3. Then \(\rho(a_j, a_k) \le \rho(a_j, z_j) + \rho(z_j, z_k) + \rho(z_k, a_k) < \frac{1}{j} + \rho(z_j, z_k) + \frac{1}{k} \to 0\), so \((a_k)\) is Cauchy in \(A\).
4.4. By hypothesis, \((a_k)\) converges to some \(z \in Z\).
4.5. Then \(\rho(z_k, z) \le \rho(z_k, a_k) + \rho(a_k, z) < \frac{1}{k} + \rho(a_k, z) \to 0\), so \(z_k \to z\), proving \(Z\) is complete.
5. Part (e): Completeness of \((Y, D)\).Proof:
5.1. Let \((h(x_k))_{k=1}^\infty\) be a Cauchy sequence in \(h(X)\).
5.2. Since \(h\) is an isometry, \(d(x_j, x_k) = D(h(x_j), h(x_k)) \to 0\), so \(\mathbf{x} = (x_1, x_2, \dots) \in \tilde{X}\).
5.3. By 3, \(h(x_k) \to [\mathbf{x}] \in Y\).
5.4. Since \(h(X)\) is dense in \(Y\) and every Cauchy sequence in \(h(X)\) converges in \(Y\), part (d) implies \((Y, D)\) is complete. Q.E.D.