Definition:Uniform Cauchy Criterion

Definition

Let $S \subseteq \mathbb R$.

Let $\sequence {f_n}$ be a sequence of real functions $S \to \R$.

We say that $\sequence {f_n}$ satisfies the uniform Cauchy criterion or is uniformly Cauchy on $S$ if for all $\varepsilon \in \R_{> 0}$, there exists $N \in \N$ such that:

$\size {\map {f_n} x - \map {f_m} x} < \varepsilon$

for all $x \in S$ and $n, m > N$.

By Sequence of Functions is Uniformly Cauchy iff Uniformly Convergent, this criterion gives a necessary and sufficient condition for a sequence of real functions to be uniformly convergent.

Source of Name

This entry was named for Augustin Louis Cauchy.

Sources

• 1973: Tom M. Apostol: Mathematical Analysis (2nd ed.) ... (previous) ... (next): $\S 9.5$: The Cauchy Condition for Uniform Convergence: Theorem $9.3$