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2: Sequences

Last updated

Sep 5, 2021
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- 1.6: Applications of the Completeness Axiom
- 2.1: Convergence

Lafferriere, Lafferriere, and Nguyen
Portland State University via PDXOpen: Open Educational Resources

( \newcommand{\kernel}{\mathrm{null}\,}\)

We introduce the notion of limit first through sequences. As mentioned in Chapter 1, a sequence is just a function with domain $N$ . More precisely, a sequence of elements of a set $A$ is a function $f : N \to A$ . We will denote the image of $n$ under the function with subscripted variables, for example, $a_{n} = f (n)$ . We will also denote sequences by ${a_{n}}_{n = 1}^{\infty}$ , ${a_{n}}_{n}$ , or even ${a_{n}}$ . Each value $a_{n}$ is called a term of the sequence, more precisely, the $n$ -th term of the sequence.

Example $2.1$

Consider the sequence $a_{n} = \frac{1}{n}$ for $n \in N$ .

Solution

This is a sequence of rational numbers. On occasion, when the pattern is clear, we may list the terms explicitly as in

$\begin{matrix} (1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \dots \end{matrix}$

Example $2.2$

Let $a_{n} = {(- 1)}^{n}$ for $n \in N$ . This is a sequence of integers, namely,

$\begin{matrix} - 1, 1, - 1, 1, - 1, 1, \dots \end{matrix}$

Solution

Note that the sequence takes on only two values. This should not be confused with the two-element set ${1, - 1}$ .

2.1: Convergence
2.2: Limit Theorems
2.3: Monotone Sequences
2.4: The Bolazno-Weierstrass Theorem
The Bolzano-Weierstrass Theorem is at the foundation of many results in analysis. it is, in fact, equivalent to the completeness axiom of the real numbers.
2.5: Limit Superior and Limit Inferior
2.6: Open Sets, Closed Sets, Compact Sets, and Limit Points

Example 2.1

Example 2.2

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Example $2.1$

Example $2.2$