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\centerline{Math 101}
\centerline{Homework 7}

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\noindent 1)   Prove that every non-empty set of integers which is bounded above has a greatest element.

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\noindent 2)  Prove that the LUB Axiom follows from the GLB Axiom.

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\noindent 3) Prove that least upper bounds are unique.  That is if $a$ and $b$ are both least upper bounds for $X$, prove that $a=b$.

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\noindent 4)  Prove that between every pair of real numbers there is an irrational number.


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\centerline{Homework 8}


\medskip
 Problem 4.16 and the following.
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  \noindent 1)   Prove that the sequence $\{(-1)^n(1-\frac{1}{n})\}$ diverges.
  
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  \noindent 2) If $\{x_n\}$ converges to 0 then $\forall c\in \mathbb{R}$, $\{cx_n\}$ converges to 0. 
  
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  \noindent 3)  Prove that $\{x_n\}$ converges to $0$ iff for every $\varepsilon >0$, $(-\varepsilon, \varepsilon)$ contains either all the terms of $\{x_n\}$ or all but finitely many terms of $\{x_n\}$.
  
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  \noindent 4)  Prove that $x_n\to \ell$ iff $\{x_n-\ell\}$ converges to zero. 
  
 
 

  

   
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  \centerline{Homework 9}

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  \noindent 1) Let $\{x_n\}$ and $\{y_n\}$ be sequences, and let $x_n\rightarrow l$.  Suppose that there exists an $M \in \mathbb{N}$ such that for all $n>M$, $y_n=x_n$.  Prove that $y_n\rightarrow l$.
  
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  \noindent 2) Suppose that $\{x_n\}$ is a sequence of integers which converges to $l$.  Prove that there exists an $N\in \mathbb{N}$ such that for all $n>N$, $x_n=l$.

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  \noindent 3)   Suppose that $z_n\rightarrow l$ and $l\neq 0$.  Prove that there is an $N \in \mathbb{N}$ such that if $n>N$ then $|z_n|>\frac{|l|}{2}$. 
  
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  \noindent 4)  Let $k\in \mathbb{N}$ and suppose that $\{x_n\}$ and $\{y_n\}$ are sequences such that $x_n\rightarrow l$ and for all $n$, $y_n=x_{n+k}$.  Prove that $y_n\rightarrow l$.

     
 
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