91Ó°ÊÓ

If \(\left(x_{n}\right) \rightarrow x\) in a metric space \(M\), show that any subsequence \(\left(x_{n_{k}}\right)\) of \(\left(x_{n}\right)\) also converges to \(x\).

Suppose that \(\left(x_{n}\right)\) is a Cauchy sequence in a metric space \(M\) and that some subsequence \(\left(x_{n k}\right)\) of \(\left(x_{n}\right)\) converges. Prove that \(\left(x_{n}\right)\) converges to the same limit as the subsequence.

Prove that if \(\left(x_{n}\right)\) is a Cauchy sequence, then the set \(\left\\{x_{n}\right\\}\) is bounded. What about the converse? Is a bounded sequence necessarily a Cauchy sequence?

Let \(\left(x_{n}\right)\) and \(\left(y_{n}\right)\) be Cauchy sequences in a metric space \(M\). Prove that the sequence \(d_{n}=d\left(x_{n}, y_{n}\right)\) converges.

Show that the space of all convergent sequences of real numbers (or complex numbers) is complete as a subspace of \(\ell^{\infty}\).

Let \(\mathcal{P}\) denote the metric space of all polynomials over \(\mathbb{C}\), with metric $$ d(p, q)=\sup _{x \in[a, b]}|p(x)-q(x)| $$ Is \(\mathcal{P}\) complete?

Prove that the metric space \(\mathbb{Z}\) of all integers, with metric \(d(n, m)=|n-m|\), is complete.

Show that the subspace \(S\) of the metric space \(C[a, b]\) (under the sup metric) consisting of all functions \(f \in C[a, b]\) for which \(f(a)=f(b)\) is complete.

If \(M \approx M^{\prime}\) and \(M\) is complete, show that \(M^{\prime}\) is also complete.

Show that the metric spaces \(C[a, b]\) and \(C[c, d]\), under the sup metric, are isometric.