2: Sequences
- Page ID
- 49105
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.
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
\[(1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \ldots \nonumber\]
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 \(\{1,-1\}\).
- 2.4: 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.