Theorems · Theorem · number theory
Real.exists_rat_eq_convergent
∀ {ξ : ℝ} {q : ℚ}, |ξ - ↑q| < 1 / (2 * ↑q.den ^ 2) → ∃ n, q = ξ.convergent nThe main result, Legendre's Theorem on rational approximation:
if ξ is a real number and q is a rational number such that |ξ - q| < 1/(2*q.den^2),
then q is a convergent of the continued fraction expansion of ξ.
This version uses Real.convergent.
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 115 from the axioms · uses propext, Classical.choice, Quot.sound
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Cited by1
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- Real.exists_convs_eq_ratproof · cited by 0