Math 2.0

statement theorem

Bolzano–Weierstrass theorem

Statement

Let (an)(a_n) be a bounded sequence of real numbers. Then (an)(a_n) has a subsequence that converges to a real number.

Remarks

The proof takes a monotone subsequence. It is bounded, so it converges by the monotone convergence theorem. The limit of the subsequence lies in each closed interval [m,M][m, M] that contains all terms, by the preservation of non-strict inequalities. The theorem holds in Rk\mathbb{R}^k for each kk. It says that each closed and bounded subset of Rk\mathbb{R}^k is sequentially compact. The theorem fails in Q\mathbb{Q}. The decimal approximations of 2\sqrt{2} are bounded, and no subsequence has a limit in Q\mathbb{Q}.

History

Bernard Bolzano proved the result in 1817, as a lemma for the intermediate value theorem. Karl Weierstrass proved it again about fifty years later and made it a central tool of analysis.

Explanations

L1 human

A bounded sequence cannot run away, so its terms must crowd together somewhere. The theorem picks infinitely many terms that approach one point of crowding. For (−1)n(-1)^n, the even terms form a subsequence that converges to 11.

What this node needs for W2
  • edge about out: the concepts that the statement is about (0 of 1)
  • edge proves in: at least one proof node (0 of 1)
  • edge instance_of or illustrates in: one example that shows the statement at work (0 of 1)
  • explanation L2 (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 10 nodes in this order. The foundations come first, and the list ends with this node.

Sequence, definition, no evidence levelSequencedefinitionReal sequence, definition, no evidence levelReal sequencedefinitionSubsequence, definition, no evidence levelSubsequencedefinitionBounded sequence, definition, no evidence levelBounded sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionMonotone sequence, definition, no evidence levelMonotone sequencedefinitionConvergent 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…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.
  • no evidence level
  1. Sequence definition
  2. Real sequence definition
  3. Subsequence definition
  4. Bounded sequence definition
  5. Limit of a sequence definition
  6. Monotone sequence definition
  7. Convergent sequence definition
  8. Monotone convergence theorem for sequences statement
  9. Every sequence has a monotone subsequence statement
  10. Bolzano–Weierstrass theorem statement

What depends on this

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

Sources

  1. Web https://en.wikipedia.org/wiki/Bolzano–Weierstrass_theorem, Proof

What links here

Used by