Chapter 3

Show that every discrete space $(S,\rho )$ is complete.

Find $sup{x}_n,inf{x}_n,max{x}_n,$ and $min{x}_n$ (if any) for sequences with general term (a) $n;$ (b) $(-1{)}^n(2-2^{2-n})$; (c) $1-\frac{2}{n};$ (d) $\frac{n(n-1)}{(n+2{)}^2}$. Which are bounded in $E^1?$

Find a unit vector in $E^4$, with positive components, that forms equal angles with the axes, i.e., with the basic unit vectors (see Problem 7 ).

Prove that if $\rho$ is a metric for $S$, then another metric ${\rho }^{\prime }$ for $S$ is given by $$\begin{gathered} \begin{array}{l}\text{ (i) }{\rho }^{\prime }(x,y)=min\\ 1,\rho (x,y) \\ \\ \text{ (ii) }{\rho }^{\prime }(x,y)=\frac{\rho (x,y)}{1+\rho (x,y)}\end{array} \end{gathered}$$ In case (i), show that globes $G_p(\epsilon )$ of radius $\epsilon \le 1$ are the same under $\rho$ and ${\rho }^{\prime }.$ In case (ii), prove that any $G_p(\epsilon )$ in $(S,\rho )$ is also a globe $G_p\left({\epsilon }^{\prime }\right)$ in $(S,{\rho }^{\prime })$ of radius $${\epsilon }^{\prime }=\frac{\epsilon }{1+\epsilon },$$ and any globe of radius ${\epsilon }^{\prime }<1$ in $(S,{\rho }^{\prime })$ is also a globe in $(S,\rho )$. (Find the converse formula for $\epsilon$ as well! $)$

Prove the additivity of the volume of intervals, namely, if $A$ is subdivided, in any manner, into $m$ mutually disjoint subintervals $A_1,A_2,\dots ,A_m$ in $E^n$, then $$vA=\sum _{i=1}^{m}v{A}_i$$ (This is true also if some $A_i$ contain common faces). [Proof outline: For $m=2$, use Problem 8 Then by induction, suppose additivity holds for any number of intervals smaller than a certain $m$ $(m>1).$ Now let $$A=\bigcup _{i=1}^m{A}_i\left(A_i\text{ disjoint }\right)$$ One of the $A_i$ (say, $A_1=[\overline{a},\overline{p}])$ must have some edge-length smaller than the corresponding edge-length of $A($ say $,{\ell }_1).$ Now cut all of $A$ into $$P=[\overline{a},\overline{d}]\text{ and }Q=A-P\text{ (Figure 4) }$$ by the plane $x_1=c(c=p_1)$ so that $A_1\subseteq P$ while $A_2\subseteq Q$. For simplicity, assume that the plane cuts each $A_i$ into two subintervals $A_i^{\prime }$ and $A_i^{\prime \prime }$. (One of them may be empty.) Then $$P=\bigcup _{i=1}^m{A}_i^{\prime }\text{ and }Q=\bigcup _{i=1}^m{A}_i^{\prime \prime }$$ Actually, however, $P$ and $Q$ are split into fewer than $m$ (nonempty) intervals since $A_1^{\prime \prime }=\mathrm{\varnothing }=A_2^{\prime }$ by construction. Thus, by our inductive assumption, $$vP=\sum _{i=1}^{m}v{A}_i^{\prime }\text{ and }vQ=\sum _{i=1}^{m}v{A}_i^{\prime \prime }$$ where $v{A}_1^{\prime \prime }=0=v{A}_2^{\prime },$ and $v{A}_i=v{A}_i^{\prime }+v{A}_i^{\prime \prime }$ by Problem $8.$ Complete the inductive proof by showing that $$vA=vP+vQ=\sum _{i=1}^{m}v{A}_i.]$$

Prove for $E^n$ that if $\overline{u}$ is orthogonal to each of the basic unit vectors ${\overline{e}}_1$, ${\overline{e}}_2,\dots ,{\overline{e}}_n,$ then $\overline{u}=\bar{0}.$ Deduce that $$\overline{u}=\bar{0}\text{ iff }(\mathrm{\forall }\overline{x}\in E^n)\overline{x}\cdot \overline{u}=0$$

Prove that if $(X,{\rho }^{\prime })$ and $(Y,{\rho }^{\prime \prime })$ are metric spaces, then a metric $\rho$ for the set $X\times Y$ is obtained by setting, for $x_1,x_2\in X$ and $y_1,y_2\in Y$, (i) $\text{Missing left or extra right}$; or (ii) $\rho ((x_1,y_1),(x_2,y_2))=\sqrt{{\rho }^{\prime }{(x_1,x_2)}^2+{\rho }^{\prime \prime }{(y_1,y_2)}^2}$

Prove the following about lines and line segments. (i) Show that any line segment in $E^n$ is a bounded set, but the entire line is not. (ii) Prove that the diameter of $L(\overline{a},\overline{b})$ and of $(\overline{a},\overline{b})$ equals $\rho (\overline{a},\overline{b})$.

Let $f:E^1\to E^1$ be given by $$f(x)=\frac{1}{x}\text{ if }x\ne 0,\text{ and }f(0)=0.$$ Show that $f$ is bounded on an interval $[a,b]$ iff $0\notin [a,b].$ Is $f$ bounded on (0,1)$?$

Prove that $$|\rho (y,z)-\rho (x,z)|\le \rho (x,y)$$ in any metric space $(S,\rho )$.