25 lines
885 B
Lean4
25 lines
885 B
Lean4
-- Library Import: Basic facts about real numbers
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import Mathlib
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open Real
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-- Definition: Function (f : ℝ → ℝ) is continuous at (x₀ : ℝ)
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def continuous_at (f : ℝ → ℝ) (x₀ : ℝ) :=
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∀ ε > 0, ∃ δ > 0, ∀ x, |x - x₀| < δ → |f x - f x₀| < ε
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-- Definition: Sequence (u : ℕ → ℝ) converges to limit (l : ℝ)
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def seq_limit (u : ℕ → ℝ) (l : ℝ) :=
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∀ ε > 0, ∃ N, ∀ n ≥ N, |u n - l| < ε
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-- In the following, f is a function, u is a sequence, and x₀ a real number
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variable
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{f : ℝ → ℝ} {u : ℕ → ℝ} {x₀ : ℝ}
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/-
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Theorem "Continuity and Limits": If the function f is continuous at x₀ and u
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is a sequence converging to x₀, then the sequence f ∘ u converges to f (x₀).
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-/
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theorem continuity_and_limits (hyp_f : continuous_at f x₀) (hyp_u : seq_limit u x₀) :
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seq_limit (f ∘ u) (f x₀) := by
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sorry
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