3.13.E: Problems on Cauchy Sequences
Without using Theorem \(4,\) prove that if \(\left\{x_{n}\right\}\) and \(\left\{y_{n}\right\}\) are Cauchy sequences in \(E^{1}(\text { or } C),\) so also are
\[
\text { (i) }\left\{x_{n}+y_{n}\right\} \quad \text { and } \quad \text { (ii) }\left\{x_{n} y_{n}\right\}.
\]
Prove that if \(\left\{x_{m}\right\}\) and \(\left\{y_{m}\right\}\) are Cauchy sequences in \((S, \rho),\) then the sequence of distances
\[
\rho\left(x_{m}, y_{m}\right), \quad m=1,2, \dots,
\]
converges in \(E^{1}\).
[Hint: Show that this sequence is Cauchy in \(E^{1} ;\) then use Theorem \(4 . ]\)
Prove that a sequence \(\left\{x_{m}\right\}\) is Cauchy in \((S, \rho)\) iff
\[
(\forall \varepsilon>0)(\exists k)(\forall m>k) \quad \rho\left(x_{m}, x_{k}\right)<\varepsilon.
\]
Two sequences \(\left\{x_{m}\right\}\) and \(\left\{y_{m}\right\}\) are called concurrent iff
\[
\rho\left(x_{m}, y_{m}\right) \rightarrow 0.
\]
Notation: \(\left\{x_{m}\right\} \approx\left\{y_{m}\right\} .\) Prove the following.
(i) If one of them is Cauchy or convergent, so is the other, and
\(\lim x_{m}=\lim y_{m}\) (if it exists).
(ii) If any two sequences converge to the same limit, they are concurrent.
Show that if \(\left\{x_{m}\right\}\) and \(\left\{y_{m}\right\}\) are Cauchy sequences in \((S, \rho),\) then
\[
\lim _{m \rightarrow \infty} \rho\left(x_{m}, y_{m}\right)
\]
does not change if \(\left\{x_{m}\right\}\) or \(\left\{y_{m}\right\}\) is replaced by a concurrent sequence (see Problems 4 and \(2 ) .\)
Call
\[
\lim _{m \rightarrow \infty} \rho\left(x_{m}, y_{m}\right)
\]
the "distance"
\[
\rho\left(\left\{x_{m}\right\},\left\{y_{m}\right\}\right)
\]
between \(\left\{x_{m}\right\}\) and \(\left\{y_{m}\right\} .\) Prove that such "distances" satisfy all metric axioms, except that \(\rho\left(\left\{x_{m}\right\},\left\{y_{m}\right\}\right)\) may be 0 even for different sequences. (When?)
Also, show that if
\[
(\forall m) \quad x_{m}=a \text { and } y_{m}=b (\text { constant }),
\]
then \(\rho\left(\left\{x_{m}\right\},\left\{y_{m}\right\}\right)=\rho(a, b)\).
Continuing Problems 4 and \(5,\) show that the concurrence relation \((\approx)\) is reflexive, symmetric, and transitive (Chapter \(1, §§4-7 ),\) i.e., an equivalence relation. That is, given \(\left\{x_{m}\right\},\left\{y_{m}\right\}\) in \(S,\) prove that
(a) \(\left\{x_{m}\right\} \approx\left\{x_{m}\right\}\) (reflexivity);
(b) if \(\left\{x_{m}\right\} \approx\left\{y_{m}\right\}\) then \(\left\{y_{m}\right\} \approx\left\{x_{m}\right\}\) (symmetry);
(c) if \(\left\{x_{m}\right\} \approx\left\{y_{m}\right\}\) and \(\left\{y_{m}\right\} \approx\left\{z_{m}\right\},\) then \(\left\{x_{m}\right\} \approx\left\{z_{m}\right\}\) (transitivity).
From Problem 4 deduce that the set of all sequences in \((S, \rho)\) splits into disjoint equivalence classes (as defined in Chapter \(1, §§4-7 )\) under the relation of concurrence \((\approx)\). Show that all sequences of one and the same class either converge to the same limit or have no limit at all, and either none of them is Cauchy or all are Cauchy.
Give examples of incomplete metric spaces possessing complete subspaces.
Prove that if a sequence \(\left\{x_{m}\right\} \subseteq(S, \rho)\) is Cauchy then it has a subsequence \(\left\{x_{m_{k}}\right\}\) such that
\[
(\forall k) \quad \rho\left(x_{m_{k}}, x_{m_{k+1}}\right)<2^{-k} .
\]
Show that every discrete space \((S, \rho)\) is complete.
Let \(C\) be the set of all Cauchy sequences in \((S, \rho) ;\) we denote them by capitals, e.g., \(X=\left\{x_{m}\right\} .\) Let
\[
X^{*}=\{Y \in C | Y \approx X\}
\]
denote the equivalence class of \(X\) under concurrence, \(\approx\) (see Problems 2, \(5^{\prime},\) and \(5^{\prime \prime}\) "). We define
\[
\sigma\left(X^{*}, Y^{*}\right)=\rho\left(\left\{x_{m}\right\},\left\{y_{m}\right\}\right)=\lim _{m \rightarrow \infty} \rho\left(x_{m}, y_{m}\right).
\]
By Problem \(5,\) this is unambiguous, for \(\rho\left(\left\{x_{m}\right\},\left\{y_{m}\right\}\right)\) does not depend on the particular choice of \(\left\{x_{m}\right\} \in X^{*}\) and \(\left\{y_{m}\right\} \in Y^{*} ;\) and \(\lim \rho\left(x_{m}, y_{m}\right)\) exists by Problem \(2 .\)
Show that \(\sigma\) is a metric for the set of all equivalence classes \(X^{*}\) \((X \in C) ;\) call this set \(C^{*} .\)
Continuing Problem \(9,\) let \(x^{*}\) denote the equivalence class of the sequence with all terms equal to \(x\) ; let \(C^{\prime}\) be the set of all such "constant" equivalence classes (it is a subset of \(C^{*} ) .\)
Show that \(C^{\prime}\) is dense in \(\left(C^{*}, \sigma\right),\) i.e., \(\overline{C^{\prime}}=C^{*}\) under the metric \(\sigma\). (See \(§ 16,\) Definition \(2 . )\)
[Hint: Fix any "point" \(X^{*} \in C^{*}\) and any globe \(G\left(X^{*} ; \varepsilon\right)\) about \(X^{*}\) in \(\left(C^{*}, \sigma\right) .\) We must show that it contains some \(x^{*} \in C^{\prime}\).
By definition, \(X^{*}\) is the equivalence class of some Cauchy sequence \(X=\left\{x_{m}\right\}\) in \((S, \rho),\) so
\[
(\exists k)(\forall m, n>k) \quad \rho\left(x_{m}, x_{n}\right)<\frac{\varepsilon}{2} .
\]
Fix some \(x=x_{n}(n>k)\) and consider the equivalence class \(x^{*}\) of the sequence \(\{x, x, \ldots, x, \ldots\} ;\) thus, \(x^{*} \in C^{\prime},\) and
\[
\sigma\left(X^{*}, x^{*}\right)=\lim _{m \rightarrow \infty} \rho\left(x_{m}, x\right) \leq \frac{\varepsilon}{2} . \quad(\mathrm{Why} ?)
\]
Thus \(x^{*} \in G\left(X^{*}, \varepsilon\right),\) as required. \(]\)
Two metric spaces \( (S, \rho ) \) and \( (T, \sigma ) \) are said to be \(isometric\) iff there is a map \(f: S \longleftrightarrow_{onto} T \) such that
\[
( \forall x, y \in S ) \quad \rho (x, y) = \sigma(f(x), f(y)).
\]
Show that the spies \( ( S, \rho ) \) and \( ( C', \sigma) \) of Problem 10 are \(isometric\). Note that it is customary not to distinguish between two isometric spaces, treating each of them as just an "isometric copy" of the other. Indeed, distances in each of them are alike.
[Hint: Define \(f(x) = x*.]
Continuing Problems 9 to \(11,\) show that the space \(\left(C^{*}, \sigma\right)\) is complete. Thus prove that for every metric space \((S, \rho),\) there is a complete metric space \( (C^{*}, \sigma )\) containing an isometric copy \(C^{\prime}\) of \(S,\) with \(C^{\prime}\) dense in \(C^{*} . C^{*}\) is called a completion of \((S, \rho)\).
[ Hint: Take a Cauchy sequence \( \{X_{m}^{*}\} \text { in } (C^{*}, \sigma ) \). By Problem 10, each globe \(G\left(X_{m}^{*} ; \frac{1}{m}\right)\) contains some \(x_{m}^{*} \in C^{\prime},\) where \(x_{m}^{*}\) is the equivalence class of
\[
\left\{x_{m}, x_{m}, \ldots, x_{m}, \ldots\right\}
\]
and \(\sigma\left(X_{m}^{*}, x_{m}^{*}\right)<\frac{1}{m} \rightarrow 0 .\) Thus by Problem \(4,\left\{x_{m}^{*}\right\}\) is Cauchy in \(\left(C^{*}, \sigma\right),\) as is \(\left\{X_{m}^{*}\right\} .\) Deduce that \(X=\left\{x_{m}\right\} \in C,\) and \(X^{*}=\lim _{m \rightarrow \infty} X_{m}^{*}\) in \(\left(C^{*}, \sigma\right) . ]\)