Chapter 1

Show that the LUB propertyimplies the monotonic-sequence property by proving the following: If \(a_{1} \leq a_{2} \leq \cdots\) is a bounded increasing real sequence, then \(\lim _{n \rightarrow x} a_{n}=L\) where \(L=\) sup \(\left\\{a_{1}, a_{2}, a_{3}, \ldots,\right.\),

\text { How would you describe the world lines of two particles that collide and destroy each other? }

Show that the sequence defined by \(p_{n}=(n, 1 / n)\) does not converge.

Give the domain of definition of each function \(f\) defined below, and describe or sketch its graph: \((a) f(x)=1 /\left(1+x^{2}\right)\) (b) \(f(x, y)=4-x^{2}-y^{2}\) (c) \(f(x)=x /(x-1)\) (d) \(f(x, y)=1 /\left(x^{2}-y^{2}\right)\) (e) \(f(x, y)=\begin{aligned} \mid 1 & \text { for } xy \end{aligned}\)

By quoting appropriate definitions and statements. verify the following assertions. Sketches may be helpful. (i) The set \(W=\\{\) all \(p=(x, y)\) with \(1 \leq|p| \leq 2\\}\) is closed. bounded, connected, and bdy \((W)\) is the disconnected set, consisting of two circles of radius 1 and 2 . (ii) The set \(R=\\{\) all \((x, y)\) with \(x \geq 0\); is closed. unbounded, connected, and its boundary 15 the vertical axis. (iii) The set \(T=\\{\) all \((x, y)\) with \(|x|=|y|\) is closed. connected, unbounded. has empty interior, and \(\operatorname{bdy}(T)=T\). (iv) The set \(Q=\\{\) all \((x, y)\) with \(x\) and \(y\) integers' is closed. unbounded. infinite. countable. disconnected. and its boundary is itself.

Fill in the missing details in the following proof that the LUB property implies that \(\mathbf{R}\) is connected. (a) Let \(\mathbf{R}=A \cup B\) where \(A\) and \(B\) are mutually separated. Then, \(A\) and \(B\) are both open. (b) With \(a_{0} \in A\) and \(b_{0} \in B\), assume \(a_{0}

Show that the sequence described by \(p_{n}=\left(\begin{array}{c}n+1 \\\ n\end{array}, \frac{(-1)^{n}}{n}\right)\) converges.

Let \(f(x)=x^{2}+x, g(x, y)=x y\), and \(h(x)=x+1 .\) What are: (a) \(f(g(1,2))\) (b) \(h(f(3))\) (c) \(g(f(1), h(2))\) (d) \(g(f(x), h(y))\) (e) \(g(h(x), f(x))\) \((f) f(g(x, h(y)))\) (g) \(f(f(x))\)

Draw a sketch to illustrate the following events: A photon vanishes, giving rise to two particles, one an electron and one a positron. The electron moves off in one direction, the positron in another. The positron strikes another electron, and the two annihilate each other, giving rise to a photon which travels off. Could this be the history of only one particle?

If \(S\) is any bounded set of real numbers, show that the numbers \(\inf (S)\) and \(\sup (S)\) belong to the closure of \(S\). 4 If \(A \subset B\) and \(B\) is a bounded set in \(n\) space, show that diam \((A) \leq \operatorname{diam}(B)\). Can equitit? occur without having \(A=B\) ?

Sketch the set of points \((x, y)\) where (a) \(|x+2 y| \leq x-y\) (b) \(\left(x^{2}-y\right)\left(x-y^{2}\right)<0\)

Let \(\left|p_{n+1}-q\right| \leq c\left|p_{n}-q\right|\) for all \(n\), where \(c<1 .\) Show that \(\lim _{n \rightarrow \infty} p_{n}=q .\)

\((a)\) If \(F(x)=x^{2}+x\) and \(G(s)=s+s^{2}\), are \(F\) and \(G\) different functions? (b) If \(F(x, y)=x^{2}+y\) and \(G(x, y)=x+y^{2}\), are \(F\) and \(G\) different functions?

\text { If } A \text { and } B \text { are sets and } A \subset B, \text { what are } A \cup B \text { and } A \cap B \text { ? }

If \(A \subset B\) and \(B\) is a bounded set in \(n\) space, show that diam \((A) \leq \operatorname{diam}(B)\). Can equit lit? occur without having \(A=B\) ?

Show that \(\left|p_{1}+p_{2}+p_{3}+\cdots+p_{n}\right| \leq\left|p_{1}\right|+\left|p_{2}\right|+\cdots+\left|p_{n}\right|\)

Let \(\left\\{p_{n}\right\\}\) and \(\left\\{q_{n}\right\\}\) be sequences in 3 -space with \(p_{n} \rightarrow p\) and \(q_{n} \rightarrow q\). Prove that \(\lim _{n \rightarrow x} p_{n} \cdot q_{n}=p \cdot q\)

\((a)\) What is the natural domain of the function \(g(x)=\sqrt{2-x} ?\) (b) What is the natural domain of the function \(f(x)=\sqrt{x-3}+\sqrt{2-x}\) ?

For any sets \(A\) and \(B\), let \(A-B\) be the set of those things which belong to \(A\) but do not belong to \(B\). What is \(A-(A-B) ?\) Is it true that \(C \cap(A-B)=(C \cap A)-(C \cap B)\) ?

Prove that \(|p-q| \geq|p|-|q|\)

Starting at the origin in the plane, draw a polygonal line as follows: Go 1 unit east, 2 units north, 3 units west, 4 units south, 5 units east, 6 units north, and so on. Find a formula for the \(n\) th vertex of this polygon.

(a) Which is larger, \([\sqrt{243 / 3}]\) or \([12 / \sqrt{5}]\) ? (b) Find a rational number between \(\sqrt{37}\) and \(\sqrt{39}\).

Let \(A\) and \(B\) be closed sets in the plane defined by: $$ \begin{aligned} &A=\\{\text { all }(x, y) \text { with } y \geq 2 ! \\ &B=\\{\text { all }(x, y) \text { with } x \geq 0 \text { and } y \leq x /(x+1)\\} (a) Find \(d=\operatorname{dist}(A, B)\). (b) Show there does not exist \(p \in A, q \in B\) with \(|p-q|=d\). (c) Find sequences \(\left\\{p_{n}\right\\}\) in \(A\) and \(\left\\{q_{n}\right\\}\) in \(B\) with \(\lim _{n \rightarrow x}\left|p_{n}-q_{n}\right|=d\). Does either sequence converge? \end{aligned} $$

Among the following sequences, some are subsequences of others; determine all those which are so related. (a) \(1,-1,1,-1, \ldots\) (b) \(1,1,-1,1,1,-1, \ldots\) (c) \(1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \ldots\) (d) \(1, \frac{1}{4}, \frac{1}{9}, \frac{1}{16}, \frac{1}{25}, \ldots\) (e) \(1,0, \frac{1}{2}, 0, \frac{1}{3}, 0, \frac{1}{4}, 0, \ldots\)

Sketch the level curves of the function described by \(f(x, y)=x^{2}-y^{2}\)

What are the cluster points for the set $$ S=\left\\{\text { all }\left(\begin{array}{ll} 1 & 1 \\ n & m \end{array}\right) \text { with } n=1,2, \ldots, m=1,2, \ldots\right. $$

Solve for \(P\) in each of the following equations: \((a)(2,1,-3)+P=(0,2,4)\) \((b)(1,-1,4)+2 P=3 P+(2,0,5)\)

Can one have two closed sets \(A\) and \(B\) which are disjoint (and not empty) and such that dist \((A, B)=0\) ?

Use the law of cosines in the plane and the properties of the norm and scalar product to verify that \(p \cdot q=|p \| q| \cos \theta\).

Exhibit a sequence having exactly three limit points. Can a sequence have an infin?e number of limit points? No limit points? Could a divergent sequence have exactly one limit point?

Sketch the level curves for \(f\) when (a) \(f(x, y)=y^{2}-x\) (b) \(f(p)=|p|-1\) (c) \(f(p)=\left\\{\begin{array}{ll}1 & \text { when }|p|<1 \\ x-y & \text { when }|p| \geq 1\end{array}\right.\)

Solve for the points \(P\) and \(Q\) if $$ \begin{aligned} 2 P+3 Q &=(0,1,2) \\ P+2 Q &=(1,-1,3) \end{aligned} $$

Show that the intersection \(\bigcap_{1}^{x} I_{n}\) of the nested sequence of intervals \(\left\\{I_{n}\right\\}\) is empty in the following cases: (a) \(I_{n}\) is the open interval \(0

Discuss the behavior of the sequence \(\left\\{a_{n}\right\\}\), where $$ a_{n}=n+1+1 / n+(-1)^{n} n $$

Sketch the level surfaces for the function \(f(x, y, z)=x^{2}+y^{2}-z^{2}\)

By constructing an example, show that the union of an infinite collection of closed sets does not have to be closed.

Solve for \(P\) and \(Q\) if $$ \begin{aligned} 3 P+Q &=(1,0,1,-4) \\ P-Q &=(2,1,2,3) \end{aligned} $$

Let \(\left\\{R_{n}\right\\}\) be a sequence of closed bounded rectangles in the plane, with \(R_{1} \supset R_{2} \supset R_{3} \supset \cdots ;\) describe \(R_{n}\) by $$ R_{n}=\left\\{\text { all }(x, y) \text { with } a_{n} \leq x \leq b_{n}, c_{n} \leq y \leq d_{n}\right\\} $$ Prove that \(\bigcap_{1}^{\alpha} R_{n} \neq \varnothing\)

Find the equation of the hyperplane in 4 -space which goes through the point \(p_{0}=\) \((0,1,-2,3)\) perpendicular to the vector \(\mathrm{a}=(4,3,1,-2)\).

If \(a_{n} \leq x_{n} \leq b_{n}\) and \(\lim _{n \rightarrow x} a_{n}=\lim _{n \rightarrow x} b_{n}=L\), show that \(\lim _{n \rightarrow x} x_{n}=L\). (This is sometimes called the "sandwich" property.)

Let \(F(x, y, z, t)=(x-t)^{2}+y^{2}+z^{2} .\) By interpreting this as the temperature at the point \((x, y, z)\) at time \(t\), see if you can get a feeling for the behavior of the function.

$$ \text { Let } A=(1,1,3) \text { and } B=(2,-1,1) \text { . Can you find a point } p \text { such that } p \cdot A=0 \text { and } p \cdot B=0 \text { ? } $$

Prove that a bounded sequence of real numbers that has exactly one limit point must be convergent. Is this still true if the sequence is unbounded?

If the angle between two hyperplanes is defined as the angle between their normals, are the hyperplanes \(3 x+2 y+4 z-2 w=5\) and \(2 x-4 y+z+w=6\) orthogonal?

Find a formula for the sequence that begins with \(1 / 2,1 / 5,1 / 10,1 / 17,1 / 26, \ldots\), and show that it converges to 0 .

Construct pictures to show that each of the following is false. (i) If \(A \subset B\). then \(\operatorname{bdy}(A) \subset \operatorname{bdy}(B)\) (ii) \(\operatorname{bd} y(S)\) is the same as the boundary of the closure of \(S\). (iii) \(\operatorname{bdy}(S)\) is the same as the boundary of the interior of \(S\). (iv) The interior of \(S\) is the same as the interior of the closure of \(S .\)

Draw a diagram to illustrate that the associative law of addition, \(p+(q+r)=(p+q)+r\), holds for the operation of addition of vectors.

Show that every noncountable set of points in the plane must have a cluster point. Must it have more than one?

Write the parametric equations of the line through \((2,3,-1,1)\) which is perpendicular to the hyperplane \(3 x+2 y-4 z+w=0\)

Prove that \(\lim _{n \rightarrow x} 1 / \sqrt{n}=0\)