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\textbf{Cauchy Sequences}
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Let $\mathbb{Q}$ denote the rational numbers and assume for the moment that all quantities mentioned below are elements of $\mathbb{Q}$.  [Note: this is just a formal assumption since we have not yet formally constructed the real numbers.]

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Definition.  Suppose that $S= \{x_i\}_{i=1}^\infty$ is a sequence of numbers.  Then $\{x_i\}_{i=1}^\infty$ is said to be a Cauchy sequence provided the following hold: If $\epsilon>0$, then there exists an integer $N_\epsilon$ so that $|x_n -x_m| < \epsilon$ for all integers $n$ and $m$ greater than $N_\epsilon$.

[Note: I use the notation $N_\epsilon$ to emphasize the fact that $N_\epsilon$ depends on $\epsilon$; but if $N_\epsilon$ is replaced with $N$, the definition is equivalent.]

Equivalently:
$$\forall(\epsilon > 0) \exists (N \in \mathbb{Z})(\forall n,m \in \mathbb{Z}(( n,m > N))(|x_n -x_m| < \epsilon)).$$

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Exercise 10.1.  Express the negation of what it means for $S= \{x_i\}_{i=1}^\infty$ to be a Cauchy sequence without using the negation $\sim$ symbol.

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Exercise 10.2. For each of the following sequences, determine if $S= \{x_n\}_{n=1}^\infty$ is a Cauchy sequence:

\begin{eqnarray*}
\mbox{a.} & x_n = & 2 \\
\mbox{b.} & x_n = & n^2 \\
\mbox{c.} & x_n = & 3n - 10 \\
\mbox{d.} & x_n = & \frac 1n \\
\mbox{e.} & x_n = & \frac {1}{7n + 3} \\
\mbox{f.} & x_n = & (-1)^n\frac 1n \\
\mbox{g.} & x_n = & (-1)^n\frac {1}{7n + 3} \\
\mbox{h.} & x_n = & \sum_{k=1}^n \frac 12 \\
\mbox{i.} & x_n = & \sum_{k=1}^n \Big(\frac 12\Big)^k \\
\mbox{j.} & x_n = & \sum_{k=1}^n \frac 1{k^2} \\
\mbox{a'.} & x_n = & (-1)^n 2 \\
\mbox{b'.} & x_n = &\frac{n-1}{n+1} \\
\mbox{b''.} & x_n = &(-1)^n \frac{n-1}{n+1} \\
\mbox{c'.} & x_n = & \frac{n-1}{5n+1} \\
\mbox{c''.} & x_n = & (-1)^n  \frac{n-1}{5n+1}
.
\end{eqnarray*}

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Definition.  A sequence$\{x_n\}_{n=1}^\infty$ is said to be bounded if there is a number $B$ so that $|x_n| < B$ for all $n$.

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Theorem 10.1 If $S= \{x_n\}_{n=1}^\infty$ is a Cauchy sequence, then $S$ is bounded.

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Theorem 10.2 If each of $S_1= \{x_n\}_{n=1}^\infty$ and $S_2 = \{y_n\}_{n=1}^\infty$ is a Cauchy sequence, then $\{x_n+ y_n\}_{n=1}^\infty$ is a Cauchy sequence.

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Theorem 10.3 If $B$ is a number and $S = \{x_n\}_{n=1}^\infty$ is a Cauchy sequence, then $\{Bx_n \}_{n=1}^\infty$ is a Cauchy sequence.

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Definition.  The sequence of rational numbers $S = \{r_i\}_{i=1}^\infty$ is said to converge to $0$ if and only if $\epsilon >0$ then there exists an integer $N_\epsilon$ so that if $n>N_\epsilon$ then
\begin{eqnarray*}
|r_n| & < & \epsilon.
\end{eqnarray*}

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Let $\hat R$ be the collection of all sequences of rational numbers.  Define the relation $\sim$ on $\hat R$ by $\{x_i\}_{i=1}^\infty \sim \{y_i\}_{i=1}^\infty$ if and only if $\{x_i - y_i\}_{i=1}^\infty$ converges to $0$.

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Theorem 10.4.  The relation $\sim$ defined on $\hat R$ is an equivalence relation.

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