Problem 1
Show that \(E^{2}\) becomes a metric space if distances \(\rho(\bar{x}, \bar{y})\) are defined by (a) \(\rho(\bar{x}, \bar{y})=\left|x_{1}-y_{1}\right|+\left|x_{2}-y_{2}\right|\) or (b) \(\rho(\bar{x}, \bar{y})=\max \left\\{\left|x_{1}-y_{1}\right|,\left|x_{2}-y_{2}\right|\right\\}\) where \(\bar{x}=\left(x_{1}, x_{2}\right)\) and \(\bar{y}=\left(y_{1}, y_{2}\right) .\) In each case, describe \(G_{\overline{0}}(1)\) and \(S_{\overline{0}}(1) .\) Do the same for the subspace of points with nonnegative coordinates.
Problem 1
Show that if a set \(A\) in a metric space is bounded, so is each subset \(B \subseteq A\).
Problem 2
Prove that if \(\left\\{x_{m}\right\\}\) and \(\left\\{y_{m}\right\\}\) are Cauchy sequences in \((S, \rho),\) then the sequence of distances $$ \rho\left(x_{m}, y_{m}\right), \quad m=1,2, \ldots $$ converges in \(E^{1}\).
Problem 2
Prove that \(z \bar{z}=|z|^{2}\). Deduce that \(z^{-1}=\bar{z} /|z|^{2}\) if \(z \neq 0 .{ }^{4}\)
Problem 3
Given \(\bar{x}=(-1,2,0,-7), \bar{y}=(0,0,-1,-2),\) and \(\bar{z}=(2,4,-3,-3)\) in \(E^{4}\), express \(\bar{x}, \bar{y},\) and \(\bar{z}\) as linear combinations of the basic unit vectors. Also, compute their absolute values, their inverses, as well as their mutual sums, differences, dot products, and distances. Are any of them orthogonal? Parallel?
Problem 3
Prove that a sequence \(\left\\{x_{m}\right\\}\) is Cauchy in \((S, \rho)\) iff $$ (\forall \varepsilon>0)(\exists k)(\forall m>k) \quad \rho\left(x_{m}, x_{k}\right)<\varepsilon $$
Problem 3
Prove that $$ \overline{z+z^{\prime}}=\bar{z}+\overline{z^{\prime}} \text { and } \overline{z z^{\prime}}=\bar{z} \cdot \overline{z^{\prime}} . $$ Hence show by induction that $$ \overline{z^{n}}=(\bar{z})^{n}, n=1,2, \ldots, \text { and } \sum_{k=1}^{n} a_{k} z^{k}=\sum_{k=1}^{n} \bar{a}_{k} \bar{z}^{k} . $$
Problem 3
Given a line \(\bar{x}=\bar{a}+t \vec{u}(\vec{u}=\bar{b}-\bar{a} \neq \overrightarrow{0})\) in \(E^{n},\) define \(f: E^{1} \rightarrow E^{n}\) by $$ f(t)=\bar{a}+t \vec{u} \text { for } t \in E^{1} $$ Show that \(L[\bar{a}, \bar{b}]\) is exactly the \(f\) -image of the interval [0,1] in \(E^{1},\) with \(f(0)=a\) and \(f(1)=b\), while \(f\left[E^{1}\right]\) is the entire line. Also show that \(f\) is one to one.
Problem 4
Let \(M\) be the set of all positive integers together with the "point" \(\infty\). Metrize \(M\) by setting \(\rho(m, n)=\left|\frac{1}{m}-\frac{1}{n}\right|,\) with the convention that \(\frac{1}{\infty}=0\). Verify the metric axioms. Describe \(G_{\infty}\left(\frac{1}{2}\right), S_{\infty}\left(\frac{1}{2}\right),\) and \(G_{1}(1) .\)
Problem 4
Define $$ e^{\theta i}=\cos \theta+i \sin \theta $$ Describe \(e^{\theta i}\) geometrically. Is \(\left|e^{\theta i}\right|=1 ?\)
