mathematics, a sequence of functions { f n } {\displaystyle \{f_{n}\}} from a set S to a metric space M is said to be uniformly Cauchy if: For all ε >...

In mathematics, a Cauchy sequence is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given...

methods. When put into the modern language, what Cauchy proved is that a uniformly convergent sequence of continuous functions has a continuous limit....

mathematical analysis, a metric space M is called complete (or a Cauchy space) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively...

test Cauchy's convergence test Cauchy–Hadamard theorem Cauchy product Cauchy's radical test Cauchy ratio test Cauchy sequence Uniformly Cauchy sequence Maclaurin–Cauchy...

true that uniformly continuous maps transform Cauchy sequences into Cauchy sequences. Each compact Hausdorff space possesses exactly one uniform structure...

progressively closer" is made rigorous by Cauchy nets or Cauchy filters, which are generalizations of Cauchy sequences, while "point x {\displaystyle x} towards...

Every uniformly continuous function is also Cauchy-continuous. Conversely, if the domain X {\displaystyle X} is totally bounded, then every Cauchy-continuous...

value). Low-discrepancy sequences are also called quasirandom sequences, due to their common use as a replacement of uniformly distributed random numbers...

analysis, a Cauchy space is a generalization of metric spaces and uniform spaces for which the notion of Cauchy convergence still makes sense. Cauchy spaces...

The Cauchy distribution, named after Augustin-Louis Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as...

analysis is the Cauchy criterion for convergence of sequences: a sequence of real numbers is convergent if and only if it is a Cauchy sequence. This remains...

for uniform spaces. Instead of working with Cauchy sequences, one works with Cauchy filters (or Cauchy nets). A Cauchy filter (respectively, a Cauchy prefilter)...

equations Cauchy–Schwarz inequality Cauchy sequence Cauchy surface Cauchy's theorem (geometry) Cauchy's theorem (group theory) Maclaurin–Cauchy test His...

In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a...

common definitions of real numbers include equivalence classes of Cauchy sequences (of rational numbers), Dedekind cuts, and infinite decimal representations...

length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well-defined limit that is within...

converges to x . {\displaystyle x.} A Cauchy net generalizes the notion of Cauchy sequence to nets defined on uniform spaces. A net x ∙ = ( x a ) a ∈ A {\displaystyle...

converges uniformly. Analogously, one can prove that ∑ k = 1 ∞ | f k ( x ) | {\displaystyle \sum _{k=1}^{\infty }|f_{k}(x)|} converges uniformly. A more...

a unique solution. It is also known as Picard's existence theorem, the Cauchy–Lipschitz theorem, or the existence and uniqueness theorem. The theorem...

m > N = max{N(ε, x1), ..., N(ε, xK)}. Consequently, the sequence {fn} is uniformly Cauchy, and therefore converges to a continuous function, as claimed...

member of a uniformly equicontinuous set of functions is uniformly continuous, and every finite set of uniformly continuous functions is uniformly equicontinuous...

comparisons to flattened-out versions of a series leads to Cauchy's condensation test: if the sequence of terms a n {\displaystyle a_{n}} is non-negative and...

metric space in which every Cauchy sequence is also convergent, that is, Cauchy sequences are equivalent to convergent sequences, is known as a complete metric...

however, a Cauchy sequence need not converge. In addition, for real-valued sequences that are monotonic, it can be shown that the sequence is bounded...

uniform on compact sets and the limit is a harmonic function on G. The theorem is a corollary of Harnack's inequality. If un(y) is a Cauchy sequence for...

Peano existence theorem, Peano theorem or Cauchy–Peano theorem, named after Giuseppe Peano and Augustin-Louis Cauchy, is a fundamental theorem which guarantees...

functions f n {\displaystyle f_{n}} are continuous and the sequence converges uniformly, by the uniform convergence theorem. This theorem can be used to show...

finite ε-net. A metric space is said to be totally bounded if every sequence admits a Cauchy subsequence; in complete metric spaces, a set is compact if and...

statistics, a wrapped Cauchy distribution is a wrapped probability distribution that results from the "wrapping" of the Cauchy distribution around the...