Definition:Null Sequence

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.