Theorems · Theorem · number theory
Real.exists_rat_abs_sub_lt_and_lt_of_irrational
∀ {ξ : ℝ}, Irrational ξ → ∀ (q : ℚ), ∃ q', |ξ - ↑q'| < 1 / ↑q'.den ^ 2 ∧ |ξ - ↑q'| < |ξ - ↑q|Given any rational approximation q to the irrational real number ξ, there is
a good rational approximation q' such that |ξ - q'| < |ξ - q|.
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 112 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites26
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Realstatement and proof · cited by 25,697
- Nat.cast_oneproof · cited by 2,501
- absstatement and proof · cited by 1,814
- zero_lt_oneproof · cited by 598
- lt_of_lt_of_leproof · cited by 438
- lt_of_le_of_ltproof · cited by 432
- mul_posproof · cited by 374
- LT.lt.transproof · cited by 370
- sqproof · cited by 280
- Nat.cast_leproof · cited by 159
- Irrationalstatement and proof · cited by 139
- sub_ne_zeroproof · cited by 119
Cited by1
Results whose statement or proof uses this declaration.
- Real.infinite_rat_abs_sub_lt_one_div_den_sq_of_irrationalproof · cited by 2