Chapter 1

Show that the set \(\mathrm{Q}\) of rational numbers \(\mathrm{x}\) such that \(0<\mathrm{x}<1\) is countably infinite.

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

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 \(\mathrm{S}\).

Let \(\mathrm{X}\) be a topological space and let \(\mathrm{C}\) and \(\mathrm{U}\) be subsets of \(\mathrm{X}\). Define \(\mathrm{C}\) to be closed if \(\mathrm{C}\) contains all its limit points and define \(\mathrm{U}\) to be open if every point \(\mathrm{p} \in \mathrm{U}\) has a neighborhood which is contained in U. Assuming these definitions show that the following statements are equivalent for a subset \(\mathrm{S}\) of \(\mathrm{X}\) i) \(\mathrm{S}\) is closed in \(\mathrm{X}\) ii) \(\mathrm{X}-\mathrm{S}\) is open in \(\mathrm{X}\) iii) \(\mathrm{S}=[\mathrm{S}]\)

Show that: (a) \({ }^{n} U_{i=1} A_{i}\) is closed if all \(\mathrm{A}_{\mathrm{i}}\) the are; b) \(\cap_{i \in \wedge} A_{i}\) is closed if all the \(\mathrm{A}_{\mathrm{i}}\) are; c) \(^{\infty} \cap_{i=1} k_{i}\) is not empty if the \(\mathrm{K}_{\mathrm{i}}\) are non empty closed intervals in \(\mathrm{R}\) and \(\mathrm{K}_{\mathrm{i}} \supseteq \mathrm{K}_{\mathrm{i}+1}\) for all \(\mathrm{i} \geq 1 .\left(\mathrm{A}_{\mathrm{i}} \subseteq \mathrm{R}^{\mathrm{m}}\right.\) for all i).

Prove: Every bounded infinite subset \(\mathrm{S}\) of \(\mathrm{R}^{\mathrm{n}}\) has a limit point.

Prove the triangle inequality in \(R^{n}\)

Show that any open spherical neighborhood (or ball) in \(\mathrm{R}^{n}\) is an open convex set.

Let \(\mathrm{T}\) be a linear transformation of \(\mathrm{R}^{\mathrm{m}}\) to \(\mathrm{R}^{\mathrm{n}}\). Show that there exists a number \(\lambda\) such that $$ \|\mathrm{TX}\| \leq \lambda \| \mathrm{X} $$ for all \(\mathrm{X} \in \mathrm{R}^{\mathrm{m}}\).

Show that the following are equivalent metrics for \(\mathrm{R}^{n}\). (i) \(d_{1}(x, y)=\|x-y\|_{1}=\left\\{^{n} \sum_{i=1}\left(x_{i}-y_{i}\right)^{2}\right\\}^{1 / 2}\) (ii) \(\mathrm{d}_{2}(\mathrm{x}, \mathrm{y})=\|\mathrm{x}-\mathrm{y}\|_{2}=\left|\mathrm{x}_{\mathrm{i}}-\mathrm{y}_{\mathrm{i}}\right|\), (iii) \(\mathrm{d}_{3}(\mathrm{x}, \mathrm{y})=\|\mathrm{x}-\mathrm{y}\|_{3}={ }^{\mathrm{n}} \sum_{\mathrm{i}=1} \mid \mathrm{x}_{\mathrm{i}}-\mathrm{y}_{\mathrm{i}}+\)

State and prove the Heine-Borel Theorem in \(R\) and show, by examples, that the conditions of the theorem are necessary.

Show that a subset of \(R^{2}\) is connected if it is path connected, but the converse is not necessarily true.

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.