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