# Definition:Limit of Sequence/Real Numbers

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## Contents

## Definition

Let $\sequence {x_n}$ be a sequence in $\R$.

Let $\sequence {x_n}$ converge to a value $l \in \R$.

Then $l$ is a **limit of $\sequence {x_n}$ as $n$ tends to infinity**.

This is usually written:

- $\displaystyle l = \lim_{n \mathop \to \infty} x_n$

## Also see

## Also known as

A **limit of $\sequence {x_n}$ as $n$ tends to infinity** can also be presented more tersely as a **limit of $\sequence {x_n}$** or even just **limit of $x_n$**.

Some sources present $\displaystyle \lim_{n \mathop \to \infty} x_n$ as $\lim_n x_n$.

## Sources

- 1962: Bert Mendelson:
*Introduction to Topology*... (previous) ... (next): $\S 2.5$: Limits: Definition $5.1$ - 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 4$: Convergent Sequences: $\S 4.4$: Definition of Convergence