Uniformly Cauchy sequence

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In mathematics, a sequence of functions { f n } {\displaystyle \{f_{n}\}} {\displaystyle \{f_{n}\}} from a set S to a metric space M is said to be uniformly Cauchy[1] if:

  • For all ε > 0 {\displaystyle \varepsilon >0} {\displaystyle \varepsilon >0}, there exists N > 0 {\displaystyle N>0} {\displaystyle N>0} such that for all x ∈ S {\displaystyle x\in S} {\displaystyle x\in S}: d ( f n ( x ) , f m ( x ) ) < ε {\displaystyle d(f_{n}(x),f_{m}(x))<\varepsilon } {\displaystyle d(f_{n}(x),f_{m}(x))<\varepsilon } whenever m , n > N {\displaystyle m,n>N} {\displaystyle m,n>N}.

Another way of saying this is that d u ( f n , f m ) → 0 {\displaystyle d_{u}(f_{n},f_{m})\to 0} {\displaystyle d_{u}(f_{n},f_{m})\to 0} as m , n → ∞ {\displaystyle m,n\to \infty } {\displaystyle m,n\to \infty }, where the uniform distance d u {\displaystyle d_{u}} {\displaystyle d_{u}} between two functions is defined by

d u ( f , g ) := sup x ∈ S d ( f ( x ) , g ( x ) ) . {\displaystyle d_{u}(f,g):=\sup _{x\in S}d(f(x),g(x)).} {\displaystyle d_{u}(f,g):=\sup _{x\in S}d(f(x),g(x)).}

Convergence criteria

A sequence of functions {fn} from S to M is pointwise Cauchy if, for each x ∈ S, the sequence {fn(x)} is a Cauchy sequence in M. This is a weaker condition than being uniformly Cauchy.

In general a sequence can be pointwise Cauchy and not pointwise convergent, or it can be uniformly Cauchy and not uniformly convergent. Nevertheless, if the metric space M is complete, then any pointwise Cauchy sequence converges pointwise to a function from S to M. Similarly, any uniformly Cauchy sequence will tend uniformly to such a function.

The uniform Cauchy property is frequently used when the S is not just a set, but a topological space, and M is a complete metric space. The following theorem holds:

  • Let S be a topological space and M a complete metric space. Then any uniformly Cauchy sequence of continuous functions fn : S → M tends uniformly to a unique continuous function f : S → M.

Generalization to uniform spaces

A sequence of functions { f n } {\displaystyle \{f_{n}\}} {\displaystyle \{f_{n}\}} from a set S to a uniform space U is said to be uniformly Cauchy if:

  • For any entourage E of U, there exists N > 0 {\displaystyle N>0} {\displaystyle N>0} such that, for all x ∈ S {\displaystyle x\in S} {\displaystyle x\in S}, ( f n ( x ) , f m ( x ) ) ∈ E {\displaystyle (f_{n}(x),f_{m}(x))\in E} {\displaystyle (f_{n}(x),f_{m}(x))\in E} whenever m , n > N {\displaystyle m,n>N} {\displaystyle m,n>N}.

See also

References