91Ó°ÊÓ

Investigate the behavior (convergence or divergence) of \(\Sigma a_{a}\) if (a) \(a_{n}=\sqrt{n+1}-\sqrt{n}\) (b) \(a_{n}=\frac{\sqrt{n+1}-\sqrt{n}}{n}\); (c) \(a_{n}=(\sqrt[n]{n}-1)^{n}\) (d) \(a_{n}=\frac{1}{1, t}, \quad\) for complex values of \(z\).

Prove that the Cauchy product of two absolutely convergent series converges absolutely.

Prove the following analogue of Theorem \(3.10(b):\) If \(\left\\{E_{a}\right\\}\) is a sequence of closed nonempty and bounded sets in a complete metric space \(X\), if \(E_{n} \supset E_{n+1}\), and if $$\lim _{n \rightarrow \infty} \operatorname{diam} E_{n}=0,$$ then \(\cap_{1}^{\infty} E_{n}\) consists of exactly one point.

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 \(X\).

Let \(X\) be the metric space whose points are the rational numbers, with the metric \(d(x, y)=|x-y| .\) What is the completion of this space? (Compare Exercise 24.)