Chapter 1: Chapter 1

Show that the set $\mathrm{A}=\\{\mathrm{x} \in \mathrm{R} \mid 0<\mathrm{x}<1\\}\( is uncountable. Conclude that \)R$ is uncountable.

Let \(R\) be a complete metric space with metric \(\rho\). Prove that every contraction mapping \(\mathrm{A}: \mathrm{R} \rightarrow \mathrm{R}\) has a unique fixed point.

Define boundedness and state the property of real numbers concerning least upper bounds (or greatest lower bounds).

If \(a>0\) and \(b>0\), show that there exists an integer \(\mathrm{n}\) such that na \(>\mathrm{b}\).

Prove the following give that the sequences \(\left\\{\mathrm{s}_{\mathrm{n}}\right\\}\) and \(\left\\{\mathrm{t}_{\mathrm{n}}\right\\}\) converge to \(\mathrm{s}\) and \(\mathrm{t}\) respectively: a) \(\lim _{n \rightarrow \infty}\left(\mathrm{s}_{n}+t_{n}\right)=s+t ;\) b) $\lim _{\mathrm{n} \rightarrow \infty} \mathrm{cs}_{\mathrm{n}}=\mathrm{cs}$, for constsant c; c) \(\lim _{n \rightarrow \infty}\left(c+s_{n}\right)=c+s\), for constant \(c\) d) $\lim _{\mathrm{n} \rightarrow \infty} \mathrm{s}_{\mathrm{n}} t_{\mathrm{n}}=$ st ; e) $\lim _{\mathrm{n} \rightarrow \infty}\left\\{1 / \mathrm{s}_{\mathrm{n}}\right\\}=\\{1 / \mathrm{s}\\}$, provided \(s_{n} \neq 0(\mathrm{n}=1,2, \ldots), s \neq 0\)

Show that a sequence of real-valued functions \(\left\\{\mathrm{f}_{\mathrm{n}}\right\\}\), defined on complete metric space \(\mathrm{X}\), is uniformly convergent if and only if for every \(\varepsilon>0\) there exists an integer \(\mathrm{N}\) such that $$ \mathrm{m}, \mathrm{n} \geq \mathrm{N}, \mathrm{t} \in \mathrm{X} $$ implies $$ \left|f_{n}(t)-f_{m}(t)\right| \leq \varepsilon . $$ (This is known as the Cauchy condition.)

Show that a sequence of real-valued functions \(\left\\{\mathrm{f}_{n}\right\\}\), defined on complete metric space \(\mathrm{X}\), is uniformly convergent if and only if for every \(\varepsilon>0\) there exists an integer \(\mathrm{N}\) such that $$ \mathrm{m}, \mathrm{n} \geq \mathrm{N}, \mathrm{t} \in \mathrm{X} $$ implies $$ \left|f_{n}(t)-f_{m}(t)\right| \leq \varepsilon $$ (This is known as the Cauchy condition.)

Show that every neighborhood of an accumulation point of a set \(\mathrm{S}\) contains infinitely many points of \(\mathrm{S}\).