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\centerline{Math 101}

 \centerline{Homework 17}

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\noindent 1)   Let $a \in (-1,0)$.  Prove that $a^n \rightarrow 0$.  
  
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  \noindent 2)  Let $c>1$.  Prove that $c^n\rightarrow \infty$.
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  \noindent 3)  Let $d<-1$.  Prove that $\{d^n\}$ diverges but does not diverge to either $\infty$ or $-\infty$.


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  \noindent 4) Let $N\in \mathbb{N}$.   Prove that $x_n\to \ell$ if and only if $x_{n+N}\rightarrow \ell$.


  
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\centerline{Homework 18}

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 \noindent 1)   Let $\{a_n\}$ be a sequence such that for every $n\in \mathbb{N}$ $|a_{n+1}-a_n|\leq \frac{1}{2^n}$.  Prove that $\{a_n\}$ is Cauchy.  Hint:  Use the following formula for the sum of a finite geometric series.  $$1+x+x^2+\dots+x^n=\frac{1-x^{n+1}}{1-x}$$


   
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  \noindent 2)    Let $\{a_n\}$ and $\{b_n\}$ be sequences such that for each $n\in \mathbb{N}$, $a_n\leq b_n$, and 
 $[a_1,b_1]\supseteq [a_2,b_2]\supseteq [a_3,b_3]\dots$.  Prove that there is a point $p \in \bigcap^\infty_{n=1} [a_n,b_n]$.  Note that a point $p$ is in $\bigcap^\infty_{n=1} [a_n,b_n]$ if and only if $\forall n\in \mathbb{N}$, $p\in [a_n,b_n]$.


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 \noindent 3) Prove that $\{x_n\}$ diverges iff for every $a\in \mathbb{R}$, there exists an $\epsilon>0$ and a subsequence $\{x_{n_k}\}$ of $\{x_n\}$ such that for all $k\in \mathbb{N}$, $|x_{n_k}-a|\geq \epsilon$.
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\noindent 4)  Suppose that $\{x_n\}$ is Cauchy.  Prove that for every $k\in \mathbb{N}$, the sequence $\{x_{n+k}-x_n\}$ is null.  Prove that the sequence $\{\sqrt{n}\}$ is a counterexample to the converse.

  
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