Math 2.0

statement theorem

Boundedness 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.

Explanations

L1 human

On a closed bounded interval, a continuous function cannot shoot off to infinity. Its graph stays between two horizontal lines.

What this node needs for W1
  • an inventory row: the row of the inventory that the node accounts for (0 of 1)

The writing levels say how complete the text of a node is. They do not measure its truth.

What this needs

To read this node, read these 15 nodes in this order. The foundations come first, and the list ends with this node.

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 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.
  • 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

What depends on this

83 nodes use this node, directly or through other nodes.

Sources

  1. Web https://en.wikipedia.org/wiki/Extreme_value_theorem, Proof of the boundedness theorem

What links here

Used by