Math 2.0

Real functions Proposition

Uniformly continuous functions map Cauchy sequences to Cauchy sequences

Builds on Uniform continuityCauchy sequence

Statement

Let D⊆ℝD \subseteq \mathbb{R} and let f ⁣:D→ℝf \colon D \to \mathbb{R} be uniformly continuous. Let (xn)(x_n) be a Cauchy sequence in DD. Then (f(xn))(f(x_n)) is a Cauchy sequence.

Remarks

Given ε>0\varepsilon > 0, take δ\delta from uniform continuity, and then NN with ∣xm−xn∣<δ|x_m - x_n| < \delta for m,n≥Nm, n \ge N. Continuity alone is not enough. The function 1/x1/x on (0,1)(0, 1) maps the Cauchy sequence 1/n1/n to the sequence nn. Mathlib states it as UniformContinuous.comp_cauchySeq.

Reading path

To understand this page from the ground up, read these 9 pages in order. The foundations come first.

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Limit point of a set, definition, no evidence levelLimit point of a setdefinitionSequence, definition, no evidence levelSequencedefinitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionReal sequence, definition, no evidence levelReal sequencedefinitionContinuity at a point, definition, no evidence levelContinuity at a pointdefinitionCauchy sequence, definition, no evidence levelCauchy sequencedefinitionEpsilon-delta definition of continuity, statement, no evidence levelEpsilon-delta definit…statementUniform continuity, definition, no evidence levelUniform continuitydefinitionUniformly continuous functions map Cauchy sequences to Cauchy sequences, statement, no evidence levelUniformly continuous …statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Limit point of a set definition
  2. Sequence definition
  3. Limit of a function definition
  4. Real sequence definition
  5. Continuity at a point definition
  6. Cauchy sequence definition
  7. Epsilon-delta definition of continuity statement
  8. Uniform continuity definition
  9. Uniformly continuous functions map Cauchy sequences to Cauchy sequences statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Uniform_continuity, Properties
About this page
Tag026FDZFRXS
URIm2://026FDZFRXS
Kindstatement, proposition
WritingW0 stub
MSC26A15
StatusThis copy of the site has no build data. Run m2 status 026FDZFRXS.
UsesUniform continuity, Cauchy sequence

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