Theorem: Section2sheet6Solutions.tendsTo_thirtyseven_mul (FormalisingMathematics2024/Solutions/Section02reals/Sheet6.lean)

State before:
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
⊢ TendsTo (fun n => 37 * a n) (37 * t)
Tactic:
intro ε hε
State after:
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
ε : ℝ
hε : ε > 0
⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |(fun n => 37 * a n) n - 37 * t| < ε

State before:
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
ε : ℝ
hε : ε > 0
⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |(fun n => 37 * a n) n - 37 * t| < ε
Tactic:
obtain ⟨X, hX⟩ := h (ε / 37) (by linarith)
State after:
case intro
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
ε : ℝ
hε : ε > 0
X : ℕ
hX : ∀ (n : ℕ), X ≤ n → |a n - t| < ε / 37
⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |(fun n => 37 * a n) n - 37 * t| < ε

State before:
case intro
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
ε : ℝ
hε : ε > 0
X : ℕ
hX : ∀ (n : ℕ), X ≤ n → |a n - t| < ε / 37
⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |(fun n => 37 * a n) n - 37 * t| < ε
Tactic:
use X
State after:
case h
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
ε : ℝ
hε : ε > 0
X : ℕ
hX : ∀ (n : ℕ), X ≤ n → |a n - t| < ε / 37
⊢ ∀ (n : ℕ), X ≤ n → |(fun n => 37 * a n) n - 37 * t| < ε

State before:
case h
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
ε : ℝ
hε : ε > 0
X : ℕ
hX : ∀ (n : ℕ), X ≤ n → |a n - t| < ε / 37
⊢ ∀ (n : ℕ), X ≤ n → |(fun n => 37 * a n) n - 37 * t| < ε
Tactic:
intro n hn
State after:
case h
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
ε : ℝ
hε : ε > 0
X : ℕ
hX : ∀ (n : ℕ), X ≤ n → |a n - t| < ε / 37
n : ℕ
hn : X ≤ n
⊢ |(fun n => 37 * a n) n - 37 * t| < ε

State before:
case h
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
ε : ℝ
hε : ε > 0
X : ℕ
hX : ∀ (n : ℕ), X ≤ n → |a n - t| < ε / 37
n : ℕ
hn : X ≤ n
⊢ |(fun n => 37 * a n) n - 37 * t| < ε
Tactic:
specialize hX n hn
State after:
case h
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
ε : ℝ
hε : ε > 0
X n : ℕ
hn : X ≤ n
hX : |a n - t| < ε / 37
⊢ |(fun n => 37 * a n) n - 37 * t| < ε

State before:
case h
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
ε : ℝ
hε : ε > 0
X n : ℕ
hn : X ≤ n
hX : |a n - t| < ε / 37
⊢ |(fun n => 37 * a n) n - 37 * t| < ε
Tactic:
simp
State after:
case h
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
ε : ℝ
hε : ε > 0
X n : ℕ
hn : X ≤ n
hX : |a n - t| < ε / 37
⊢ |37 * a n - 37 * t| < ε

State before:
case h
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
ε : ℝ
hε : ε > 0
X n : ℕ
hn : X ≤ n
hX : |a n - t| < ε / 37
⊢ |37 * a n - 37 * t| < ε
Tactic:
rw [← mul_sub, abs_mul, abs_of_nonneg] <;> linarith
State after:
no goals

State before:
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
ε : ℝ
hε : ε > 0
⊢ ε / 37 > 0
Tactic:
linarith
State after:
no goals