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\centerline{MATH 242 -- Principles of Analysis}
\centerline{Problem Set 2 -- due:  Feb. 7}
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\noindent
{\it `A` Section}
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\item{1.}  Let $x \in [-1,2]$.  Determine the largest and smallest
values of $|x - 5|$, $|x + 5|$.
\item{2.} Use the binomial theorem (Theorem 1.4.1) for all parts of this problem.
\itemitem{a.}  Expand using the binomial theorem and simplify as much as possible:
$$(a^2 + 3b)^5.$$
\itemitem{b.}  What is the coefficient of $x^3$ in the expansion of
$$\left({x^5 + {3\over x^2}}\right)^3.$$
\itemitem{c.}  Express with no summation: $\sum_{k=0}^n {n\choose k} 2^k$?  Explain.
\itemitem{d.}  Express with no summation: $\sum_{k=0}^n (-1)^k {n \choose k}$?  Explain.
\item{3.}  For each of the following statements, say whether the statement is true or false.
If it is false, give a counterexample; if it is true, give a short reason.
\itemitem{a.}  A set $A \subset {\bf R}$ is bounded if there exists some $B>0$ such
that $|x| \le B$ for all $x \in A$.
\itemitem{b.}  If $A, B\subset {\bf R}$ are bounded, then $A \cap B$ is also bounded.
\itemitem{c.}  If $A, B\subset {\bf R}$ are bounded, then $D = \{x + y \mid x \in A, y \in B\}$
is also bounded.
\itemitem{d.}  If $A, B\subset {\bf R}_{>0}$ are bounded, then $Q = \{y/x \mid x \in A, y \in B\}$
is also bounded.
\item{4.}
\itemitem{a.}  Let $A = [0,3) \cap (2,5]$.  What is $a = {\rm lub}\ A$?  What is $b = {\rm glb}\ A$?
Are $a,b \in A$?
\itemitem{b.}  Let $B = \{x \in {\bf R} \mid 0 < x^2 - 2x + 1 < 1\}$.  What is $a = {\rm lub}\ B$?  What is $b = {\rm glb}\ B$?   Are $a,b \in B$?
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\noindent
{\it `B` Section}
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\item{1.}  Let $x,y$ be any real numbers.
\itemitem{a.}  Show that $|x| - |y| \le |x - y|$ and deduce that $||x| - |y|| \le |x - y|$.
\itemitem{b.}  Show that if $x,y > 0$, then $x < y$ is equivalent to $x^n < y^n$ for all $n \ge 1$.
\itemitem{c.}  Show that if $0 < x < y$, then $\sqrt{y} - \sqrt{x} < \sqrt{y - x}$.
\item{2.}  Let $a,b$ be any real numbers.  Define $\max(a,b)$ and $\min(a,b)$
to be the larger and smaller of the two numbers, respectively.  (That is, $\max(a,b) = a$
if $a \ge b$ and $\max(a,b) = b$ if $b \ge a$.  Similarly for the minimum.)
Show that
$$\max(a,b) = {a + b\over 2} + {|a - b|\over 2}$$
and
$$\min(a,b) = {a + b\over 2} - {|a - b|\over 2}.$$
(Hint:  Consider the positions of $a,b$ on the number line and then ``think geometrically.'')
\item{3.}  Show by mathematical induction that
$$(1+x)^n \ge 1 + nx$$
for all $x > -1$ and all $n\ge 0$.
\item{4.}  Find a suitable $n_0$ and then show by mathematical induction
that $n! \ge 5^n$ for all $n \ge n_0$.
\bye
