# Definition:Range of Sequence

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## Definition

Let $\sequence {x_n}_{n \mathop \in A}$ be a sequence.

The **range of $\sequence {x_n}$** is the set:

- $\set {x_n: n \mathop \in A}$

## Also known as

Some treatments of this subject refer to the **range** of a sequence as the **associated set** of the sequence.

Some treatments do not bother to give it a name at all, merely referring to it as the **set of its elements**.

In keeping with the naming convention on this site it would make sense to refer to this object as the **image of (the sequence) $\sequence {x_n}$**.

However, this is rarely seen in the published literature.

## Sources

- 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $1$: Review of some real analysis: $\S 1.2$: Real Sequences - 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 4$: Convergent Sequences: $\S 4.2$: Sequences - 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $1$: General Introduction: Limit Points