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{\large Math 131 --- Homework 3 }

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%PART ALPHA.

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{\bf READ Problem D.}
Let $u$ be an upper bound of non-empty set $A$ in $\R$.
Prove that $u$ is the supremum of $A$
if and only if for all $\epsilon > 0$ there is an $a \in A$ such that
$u-\epsilon < a$.

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%(We discussed this fact in class but now I am asking you to prove it!

Note that to show that ``$S$ if and only if $T$'' you must show
that $S$ implies $T$, and $T$ implies $S$.)

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{\bf READ Problem E.} 
Let $A, B$ be nonempty subsets of $\R$ that are bounded above, 
and let $A+B=\{ a+b : a\in A, b\in
B\}$.  Show that
$$
\sup (A+B) = \sup A + \sup B.
$$

 
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{\bf Problem F.} 
Let $A, B$ be nonempty subsets {\em of positive real numbers} that are bounded above, 
and let $A\cdot B=\{ a b : a\in A, b\in B\}$.  
Show that
$$
\sup (A\cdot B) = \sup A \sup B.
$$

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{\bf Problem G.} 
(a) Let $A$ be a nonempty subset of $\R$ and suppose that $s = \sup A$ belongs to
$A$.  If $b$ is not in $A$, show that $\sup (A \cup \{ b\} )$ is equal
to the larger of the two numbers $s$ and $b$.

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(b) Use this to show that a nonempty finite set $A$ contains its
supremum.  [Hint--- use induction: show it is true first for a
  one-element set, then show that {\em if} it is true for an $n$-element set
  then it must be true for an $(n+1)$-element set.]

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%Do also {\bf Chapter 1 ( 12 )}.


%PART BETA.


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Do also {\bf Chapter 1 ( 6ab, 6cd, 12, R13, 15 )}.

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Comment:  When you are asked a question, e.g., problem 1.15, you
should always give justification.

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