Math 2.0

Real functions Theorem

Boundedness theorem

Builds on Continuous functionBolzano–Weierstrass theorem

Statement

Let a≤ba \le b and let f ⁣:[a,b]→ℝf \colon [a, b] \to \mathbb{R} be continuous. Then ff is bounded. That is, there is KK with ∣f(x)∣≤K|f(x)| \le K for all x∈[a,b]x \in [a, b].

Remarks

Suppose not. Then there are points xn∈[a,b]x_n \in [a, b] with ∣f(xn)∣>n|f(x_n)| > n. By the Bolzano–Weierstrass theorem, a subsequence converges to some c∈[a,b]c \in [a, b]. Continuity at cc gives a contradiction. The interval must be closed and bounded. The function 1/x1/x on (0,1](0, 1] is continuous but not bounded. Mathlib states a form for compact sets, as IsCompact.bddAbove_image.

Reading path

To understand this page from the ground up, read these 15 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 sequencedefinitionSubsequence, definition, no evidence levelSubsequencedefinitionContinuity at a point, definition, no evidence levelContinuity at a pointdefinitionBounded sequence, definition, no evidence levelBounded sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionMonotone sequence, definition, no evidence levelMonotone sequencedefinitionContinuous function, definition, no evidence levelContinuous functiondefinitionConvergent sequence, definition, no evidence levelConvergent sequencedefinitionMonotone convergence theorem for sequences, statement, no evidence levelMonotone convergence …statementEvery sequence has a monotone subsequence, statement, no evidence levelEvery sequence has a …statementBolzano–Weierstrass theorem, statement, no evidence levelBolzano–Weierstrass t…statementBoundedness theorem, statement, no evidence levelBoundedness theoremstatement
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. Subsequence definition
  6. Continuity at a point definition
  7. Bounded sequence definition
  8. Limit of a sequence definition
  9. Monotone sequence definition
  10. Continuous function definition
  11. Convergent sequence definition
  12. Monotone convergence theorem for sequences statement
  13. Every sequence has a monotone subsequence statement
  14. Bolzano–Weierstrass theorem statement
  15. Boundedness theorem statement

Where this leads

122 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Extreme_value_theorem, Proof of the boundedness theorem
About this page
Tag0ATVXC3NCC
URIm2://0ATVXC3NCC
Kindstatement, theorem
WritingW0 stub
MSC26A15
StatusThis copy of the site has no build data. Run m2 status 0ATVXC3NCC.
UsesContinuous function, Bolzano–Weierstrass theorem

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