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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

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We introduce the notion of limit first through sequences. As mentioned in Chapter 1, a sequence is just a function with domain $\mathbb{N}$ . More precisely, a sequence of elements of a set $A$ is a function $f: \mathbb{N} \rightarrow A$ . 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 $\left\{a_{n}\right\}_{n=1}^{\infty}$ , $\left\{a_{n}\right\}_{n}$ , or even $\left\{a_{n}\right\}$ . Each value $a_{n}$ is called a term of the sequence, more precisely, the $n$ -th term of the sequence.

Example $\PageIndex{1}$

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

Solution

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

$(1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \ldots \nonumber$

Example $\PageIndex{2}$

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

$-1,1,-1,1,-1,1, \ldots \nonumber$

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\PageIndex{1}

Example 2.2\PageIndex{2}

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Example $\PageIndex{1}$

Example $\PageIndex{2}$