Math 2.0

Real functions Proof

Proof that uniformly continuous functions map Cauchy sequences to Cauchy sequences

Builds on Uniform continuityCauchy sequence

Proof

  1. Let ε>0\varepsilon > 0.
  2. Since ff is uniformly continuous, there is δ>0\delta > 0 with ∣f(x)−f(y)∣<ε\abs{f(x) - f(y)} < \varepsilon for all x,y∈Dx, y \in D with ∣x−y∣<δ\abs{x - y} < \delta.
  3. Since (xn)(x_n) is a Cauchy sequence, there is NN with ∣xm−xn∣<δ\abs{x_m - x_n} < \delta for all m,n≥Nm, n \ge N.
  4. Let m,n≥Nm, n \ge N. The points xmx_m and xnx_n lie in DD and satisfy ∣xm−xn∣<δ\abs{x_m - x_n} < \delta. So step 2 gives ∣f(xm)−f(xn)∣<ε\abs{f(x_m) - f(x_n)} < \varepsilon.
  5. Since ε>0\varepsilon > 0 was arbitrary, (f(xn))(f(x_n)) is a Cauchy sequence.

Idea

Uniform continuity turns "inputs within δ\delta" into "outputs within ε\varepsilon", with one δ\delta for all points. A Cauchy sequence eventually has all its terms within δ\delta of each other. So the images are eventually within ε\varepsilon of each other.

Reading path

To understand this page from the ground up, read these 19 pages in order. The foundations come first. The path takes the proof Proof of the epsilon-delta form of continuity.

Show the path as a graph
Construction of the reals from Cauchy sequences, concept, no evidence levelConstruction of the r…conceptIntegers, object, no evidence levelIntegersobjectLimit point of a set, definition, no evidence levelLimit point of a setdefinitionOrdered field, definition, no evidence levelOrdered fielddefinitionRational numbers, object, no evidence levelRational numbersobjectSequence, definition, no evidence levelSequencedefinitionRing of integers, statement, no evidence levelRing of integersstatementLimit 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 sequencedefinitionThe rationals form a field, statement, no evidence levelThe rationals form a …statementThe rationals form an ordered field, statement, no evidence levelThe rationals form an…statementOrder on the reals, definition, no evidence levelOrder on the realsdefinitionInterval, definition, no evidence levelIntervaldefinitionProof of the epsilon-delta form of continuity, proof, no evidence levelProof of the epsilon-…proofEpsilon-delta definition of continuity, statement, no evidence levelEpsilon-delta definit…statementUniform continuity, definition, no evidence levelUniform continuitydefinitionProof that uniformly continuous functions map Cauchy sequences to Cauchy sequences, proof, no evidence levelProof that uniformly …proof
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. Construction of the reals from Cauchy sequences concept
  2. Integers object
  3. Limit point of a set definition
  4. Ordered field definition
  5. Rational numbers object
  6. Sequence definition
  7. Ring of integers statement
  8. Limit of a function definition
  9. Real sequence definition
  10. Continuity at a point definition
  11. Cauchy sequence definition
  12. The rationals form a field statement
  13. The rationals form an ordered field statement
  14. Order on the reals definition
  15. Interval definition
  16. Proof of the epsilon-delta form of continuity proof
  17. Epsilon-delta definition of continuity statement
  18. Uniform continuity definition
  19. Proof that uniformly continuous functions map Cauchy sequences to Cauchy sequences proof

Where this leads

1 page builds on this one.

Further reading

  1. Web https://www.jirka.org/ra/html/sec_unifcont.html, Lemma 3.4.5
About this page
Tag19VV6JCYN4
URIm2://19VV6JCYN4
Kindproof
WritingW0 stub
MSC26A15
ProvesUniformly continuous functions map Cauchy sequences to Cauchy sequences
UsesUniform continuity, Cauchy sequence
Changed4 November 2026

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