# Definition:Null Sequence

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## Contents

## Definition

### Normed Division Ring

Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring with zero $0_R$.

Let $\sequence {x_n}$ be a sequence in $R$ which converges to the limit $0_R$:

- $\displaystyle \lim_{n \mathop \to \infty} x_n = 0_R$

Then $\sequence {x_n}$ is called a **null sequence**.

## Standard Number Fields

### Complex Numbers

Let $\sequence {z_n}$ be a sequence in $\C$ which converges to a limit of $0$:

- $\displaystyle \lim_{n \mathop \to \infty} z_n = 0$

Then $\sequence {z_n}$ is called a **(complex) null sequence**.

### Real Numbers

Let $\sequence {x_n}$ be a sequence in $\R$ which converges to a limit of $0$:

- $\displaystyle \lim_{n \mathop \to \infty} x_n = 0$

Then $\sequence {x_n}$ is called a **(real) null sequence**.

### Rational Numbers

Let $\sequence {x_n}$ be a sequence in $\Q$ which converges to a limit of $0$:

- $\displaystyle \lim_{n \mathop \to \infty} x_n = 0$

Then $\sequence {x_n}$ is called a **(rational) null sequence**.

## Sources

- 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**null sequence**