lemma 7.93 A polynomial has at most $n$ roots

open in the book · parts/02-mathematical-methods/05-real-analysis.tex:2786 · p. 240

Rests on

depends_on theorem 7.35 Mean value theorem ¶
- depends_on proposition 7.29 Linearity ¶
  - depends_on definition 7.11 Sum of functions ¶
    - depends_on definition 7.9 Real function ¶
      - depends_on definition 3.43 Map ¶
  - depends_on equation 7.14 eq:ana-derivh ¶
  - depends_on proposition 7.6 Algebra of limits ¶
    - depends_on definition 7.4 Convergence ¶
      - depends_on definition 7.2 Absolute value ¶
    - depends_on proposition 7.3 Triangle inequality ¶
      - depends_on definition 7.2 Absolute value ¶ ↺
      - proves proof ch:05-real-analysis@proof-1 ¶
    - proves proof ch:05-real-analysis@proof-3 ¶
  - proves proof ch:05-real-analysis@proof-13 ¶
- depends_on theorem 7.34 Rolle ¶
  - depends_on lemma 7.33 Fermat: interior extremum ¶
    - depends_on definition 7.26 Derivative of a function at a point ¶
      - depends_on definition 7.16 Limit ¶
      - depends_on definition 7.9 Real function ¶ ↺
    - depends_on proposition 7.18 Two-sided limit from one-sided limits ¶
      - depends_on definition 7.16 Limit ¶ ↺
      - depends_on equation 7.7 eq:ana-limit-left ¶
      - depends_on equation 7.5 eq:ana-limit-right ¶
      - proves proof ch:05-real-analysis@proof-6 ¶
    - proves proof ch:05-real-analysis@proof-17 ¶
  - depends_on theorem 7.24 Extreme value theorem ¶
    - depends_on axiom 7.1 Completeness of $\R$ ¶
    - depends_on proposition 7.22 Sequential characterization ¶
      - depends_on definition 7.20 Continuity at a point ¶
      - depends_on definition 7.4 Convergence ¶ ↺
      - proves proof ch:05-real-analysis@proof-7 ¶
    - depends_on theorem 7.7 Bolzano–Weierstrass ¶
      - depends_on corollary A.45 Archimedean property and density of $\Q$ ¶
      - depends_on corollary A.46 Monotone convergence ¶
      - proves proof ch:05-real-analysis@proof-4 ¶
    - proves proof ch:05-real-analysis@proof-9 ¶
  - proves proof ch:05-real-analysis@proof-18 ¶
- proves proof ch:05-real-analysis@proof-19 ¶
depends_on theorem 7.51 Termwise differentiation ¶
- depends_on proposition 7.47 Comparison; absolute convergence ¶
  - depends_on corollary A.46 Monotone convergence ¶ ↺
  - depends_on definition 7.45 Series ¶
    - depends_on definition 7.2 Absolute value ¶ ↺
    - depends_on definition 7.4 Convergence ¶ ↺
  - depends_on theorem 7.8 Cauchy criterion ¶
    - depends_on corollary A.47 Cauchy completeness ¶
      - depends_on corollary A.46 Monotone convergence ¶ ↺
      - depends_on theorem A.40 Least-upper-bound property ¶
      - proves proof app:A-long-proofs@proof-34 ¶
    - depends_on definition 7.4 Convergence ¶ ↺
    - proves proof ch:05-real-analysis@proof-5 ¶
  - proves proof ch:05-real-analysis@proof-28 ¶
- depends_on proposition 7.27 Differentiable implies continuous ¶
  - depends_on definition 7.20 Continuity at a point ¶ ↺
  - depends_on definition 7.26 Derivative of a function at a point ¶ ↺
  - depends_on proposition 7.6 Algebra of limits ¶ ↺
  - proves proof ch:05-real-analysis@proof-11 ¶
- depends_on proposition 7.48 Ratio test ¶
  - depends_on proposition 7.47 Comparison; absolute convergence ¶ ↺
  - depends_on proposition 7.46 Geometric series ¶
    - depends_on corollary A.46 Monotone convergence ¶ ↺
    - depends_on definition 7.45 Series ¶ ↺
    - proves proof ch:05-real-analysis@proof-27 ¶
  - depends_on theorem 7.8 Cauchy criterion ¶ ↺
  - proves proof ch:05-real-analysis@proof-29 ¶
- depends_on proposition 7.6 Algebra of limits ¶ ↺
- depends_on theorem 7.50 Power series; radius of convergence ¶
  - depends_on axiom 7.1 Completeness of $\R$ ¶ ↺
  - depends_on proposition 7.47 Comparison; absolute convergence ¶ ↺
  - depends_on proposition 7.46 Geometric series ¶ ↺
  - proves proof ch:05-real-analysis@proof-31 ¶
- proves proof ch:05-real-analysis@proof-32 ¶
proves proof ch:05-real-analysis@proof-61 ¶

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type	direction	node	provenance	where
`depends_on`	→	Mean value theorem	declared	parts/02-mathematical-methods/05-real-analysis.tex:2790
`depends_on`	→	Termwise differentiation	declared	parts/02-mathematical-methods/05-real-analysis.tex:2790
`proves`	←	ch:05-real-analysis@proof-61	declared	parts/02-mathematical-methods/05-real-analysis.tex:2793