# Definition:Supremum Metric/Bounded Real Sequences

## Definition

Let $A$ be the set of all bounded real sequences.

Let $d: A \times A \to \R$ be the function defined as:

$\ds \forall \sequence {x_i}, \sequence {y_i} \in A: \map d {\sequence {x_i}, \sequence {y_i} } := \sup_{n \mathop \in \N} \size {x_n - y_n}$

where $\sup$ denotes the supremum.

$d$ is known as the supremum metric on $A$.

## Also known as

This metric is also known as the sup metric or the uniform metric.