Prove that if the sequence \(a\) converges to \(L\), then \[ (∃ N)(∀ n ≥ N)[aₙ ≥ L - 1] \]
To do so, complete the following Lean 4 theory:
import Mathlib.Data.Real.Basic import Mathlib.Tactic variable {a : ℕ → ℝ} variable {L : ℝ} def SeqLim (a : ℕ → ℝ) (L : ℝ) : Prop := ∀ ε > 0, ∃ N : ℕ, ∀ n ≥ N, |a n - L| < ε example (ha : SeqLim a L) : ∃ N, ∀ n ≥ N, a n ≥ L - 1 := by sorry
Prove that if \(aₙ\) converges to \(L\) and \(bₙ\) to \(M\), then \(aₙ+bₙ\) converges to \(L+M\).
To do so, complete the following Lean 4 theory:
import Mathlib.Data.Real.Basic import Mathlib.Tactic variable {a b c : ℕ → ℝ} variable {L M : ℝ} def SeqLim (a : ℕ → ℝ) (L : ℝ) : Prop := ∀ ε > 0, ∃ N : ℕ, ∀ n ≥ N, |a n - L| < ε example (ha : SeqLim a L) (hb : SeqLim b M) : SeqLim (a + b) (L + M) := by sorry
Prove that there are infinitely many prime numbers.
To do so, complete the following Lean 4 theory:
import Mathlib.Tactic import Mathlib.Data.Nat.Prime.Defs open Nat example (n : ℕ) : ∃ p, n ≤ p ∧ Nat.Prime p := by sorry
Prove that the sequence defined by 1, -1, 1, -1, ... does not converge.
To do so, the following proof in Lean 4 is to be completed:
import Mathlib.Data.Real.Basic import Mathlib.Tactic variable {a : ℕ → ℝ} def SeqLim (a : ℕ → ℝ) (L : ℝ) : Prop := ∀ ε > 0, ∃ N : ℕ, ∀ n ≥ N, |a n - L| < ε def SeqConv (a : ℕ → ℝ) : Prop := ∃ L, SeqLim a L example (ha : ∀ n, a n = (-1) ^ n) : ¬ SeqConv a := by sorry
In Lean, a sequence \(a₀, a₁, a₂,...\) can be represented by a function \(a : ℕ → ℝ\) so that \(a(n)\) is \(aₙ\).
The statement that \(L\) is the limit of the sequence \(a\) is defined by
def SeqLim (a : ℕ → ℝ) (L : ℝ) : Prop := ∀ ε > 0, ∃ k : ℕ, ∀ n ≥ k, |a n - L| < ε
Prove that if \(aₙ = 1/n\) for all \(n\), then the sequence \(a\) converges to 0.
To do so, complete the following Lean 4 theory:
import Mathlib.Data.Real.Basic import Mathlib.Tactic variable (a : ℕ → ℝ) def SeqLim (a : ℕ → ℝ) (L : ℝ) : Prop := ∀ ε > 0, ∃ k : ℕ, ∀ n ≥ k, |a n - L| < ε example (ha : ∀ n, a n = 1 / n) : SeqLim a 0 := by sorry
I have started publishing the series Mathematical challenges in Lean 4.
The format is simple: each week I will post a mathematical problem so that anyone interested can share their solutions in Lean 4.
At the end of the week I will publish the solutions to the challenge, which will follow this scheme:
grind.norm_num, ring, positivity or linarith.Through this process of successive refinement, the aim is for the final proof to correspond, as far as possible, to the one written in natural language.