Math 2.0

Sequences, series, summability Counterexample

x^n converges pointwise but not uniformly on [0, 1]

Builds on Pointwise convergenceUniform convergenceUniform limit theoremGeometric sequence

Counterexample

Let fn(x)=xnf_n(x) = x^n on [0,1][0, 1]. Let f(x)=0f(x) = 0 for 0≤x<10 \le x < 1 and f(1)=1f(1) = 1. For 0≤x<10 \le x < 1, xn→0x^n \to 0 by the geometric sequence limit. So fn→ff_n \to f pointwise on [0,1][0, 1]. But sup⁡0≤x<1∣xn−0∣=1\sup_{0 \le x < 1} |x^n - 0| = 1 for every nn. So the convergence is not uniform.

Remarks

Each fnf_n is continuous, but ff is not. This also shows the failure, by the uniform limit theorem. On [0,r][0, r] with 0<r<10 < r < 1, the convergence is uniform, since sup⁡[0,r]xn=rn→0\sup_{[0, r]} x^n = r^n \to 0.

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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 pointdefinitionGeometric sequence, definition, no evidence levelGeometric sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionPointwise convergence, definition, no evidence levelPointwise convergencedefinitionUniform convergence, definition, no evidence levelUniform convergencedefinitionUniform convergence on a set, definition, no evidence levelUniform convergence o…definitionUniform limit theorem, statement, no evidence levelUniform limit theoremstatementx^n converges pointwise but not uniformly on [0, 1], counterexample, no evidence levelx^n converges pointwi…counterexample
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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. Geometric sequence definition
  7. Limit of a sequence definition
  8. Pointwise convergence definition
  9. Uniform convergence definition
  10. Uniform convergence on a set definition
  11. Uniform limit theorem statement
  12. x^n converges pointwise but not uniformly on [0, 1] counterexample

Further reading

  1. Web https://en.wikipedia.org/wiki/Uniform_convergence, Examples
About this page
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Kindcounterexample
WritingW0 stub
MSC40A30
UsesPointwise convergence, Uniform convergence, Uniform limit theorem, Geometric sequence

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