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\centerline{Math 101}

\centerline{Homework 4}

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   \noindent 1)  Let $a$ and $b$ be positive rational numbers such that $\sqrt{ab}$ is irrational.  Prove that $\sqrt{a}+\sqrt{b}$ is also irrational.

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 \noindent 2)   Suppose $A$ is a non-empty set of reals and $B=\{-a|a \in A\}$.  Prove that if $A$ is bounded below, then $B$ is bounded above.
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  \noindent 3)  Let $A$ is a non-empty set of reals and $p=\mathrm{lub}(A)$, and let $B=\{-a|a \in A\}$.  Prove that $-p=\mathrm{glb}(B)$. 
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  \noindent 4)  Suppose $X$ is a set of reals such that for every $x\in X$, $a\leq x\leq b$.  Prove that there exists a positive real number $c$ such that for every $x\in X$, $|x|\leq c$.
  
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\centerline{Homework 5}

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 \noindent Problem 4.6 and the following problems.
 
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 \noindent 1)  Let $x>0$.  Prove that $\exists n\in \mathbb{N}$ st $n-1\leq x<n$.


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 \noindent 2) Let $a$ be an upper bound for a set $X$.  Prove that $a =\mathrm{lub}(X)$ iff for every $\varepsilon>0$ there is an element of $X$ in $[a-\varepsilon,a]$.    
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  \noindent 3)   Let $A=\{\frac{1}{n}|n\in \mathbb{N}\}$.  Find $\mathrm{lub}(A)$ and $\mathrm{glb}(A)$ and prove your answers.  
  
  
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  \noindent 4)  Let $A$ be a non-empty bounded 
subset of the reals, and let $B$ be the set of 
all upper bounds for $A$.  Prove that 
$\mathrm{lub}(A)=\mathrm{glb}(B)$ 

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\centerline{Homework 6}

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  \noindent Problems 4.8 (note in part (a), where it says observe, it means prove), 4.10, 4.14a), and 4.15.  
 

 
 
 
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