Math 2.0

statement proposition

Uniformly continuous functions map Cauchy sequences to Cauchy sequences

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.

Explanations

L1 human

Points of a Cauchy sequence crowd closer and closer together. A uniformly continuous function keeps their images crowding together too.

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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 node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
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  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

Sources

  1. Web https://en.wikipedia.org/wiki/Uniform_continuity, Properties

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