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Def/Sequence
From SlugmathWiki
Definition of Sequence: Suppose that $X$ is a set. Suppose that $I$ is a set of natural numbers, or more generally, of integers. Suppose, moreover, that $I$ has the following property:
- $I$ is an interval, i.e., if $a,b \in I$, and $c \in \ZZ$, and $a < c< b$, then $c \in I$.
A sequence with entries in $X$, indexed by $I$, is a function $x$ from $I$ to $X$. However, instead of traditional function notation, one uses the notation $(x_i)$ or $(x_i)_{i \in I}$ for a sequence of elements of $X$, indexed by $I$.
If $I$ is a finite set, we say that the sequence $(x_i)_{i \in I}$ is a finite sequence of elements of $X$, indexed by $I$. We discuss finite sequences separately.
Logical Connections
This definition logically relies on the following definitions and statements: Def/Natural number
The following statements and definitions logically rely on the material of this page: Def/Arithmetic progression, Def/Euclidean algorithm, Def/Generate (group), Def/Limit (sequence), Def/Quadratic sequence, Def/Sequence of differences, and Def/Sequence of primes
To visualize the logical connections between this definition and other items of mathematical knowledge, you can visit any of the following clusters, and click the "Visualize" tab: Clust/Sequences
