Theorems · Theorem · number theory
Real.exists_int_int_abs_mul_sub_le
- 1000+ list: Dirichlet's approximation theorem
∀ (ξ : ℝ) {n : ℕ}, 0 < n → ∃ j k, 0 < k ∧ k ≤ ↑n ∧ |↑k * ξ - ↑j| ≤ 1 / (↑n + 1)Dirichlet's approximation theorem:
For any real number ξ and positive natural n, there are integers j and k,
with 0 < k ≤ n and |k*ξ - j| ≤ 1/(n+1).
See also Real.exists_nat_abs_mul_sub_round_le.
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 110 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites55
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
- Finsetproof · cited by 13,712
- AddGroupproof · cited by 4,410
- one_mulproof · cited by 2,841
- Nat.cast_oneproof · cited by 2,501
- Finset.cardproof · cited by 2,327
- mul_commproof · cited by 2,262
- LT.lt.leproof · cited by 2,189
- MulZeroClass.mul_zeroproof · cited by 2,091
- Nat.cast_zeroproof · cited by 1,870
- absstatement and proof · cited by 1,814
- MulZeroClass.zero_mulproof · cited by 1,625
Cited by2
Results whose statement or proof uses this declaration.
- Real.exists_rat_abs_sub_le_and_den_leproof · cited by 1
- Real.exists_nat_abs_mul_sub_round_leproof · cited by 0