-- Mathlib/Analysis/SpecificLimits/Basic.lean
theorem tendsto_nat_ceil_mul_div_atTop {a : R} (ha : 0 ≤ a) :
    Tendsto (fun x ↦ (⌈a * x⌉₊ : R) / x) atTop (𝓝 a)

Proof state:
R : Type u_4
inst✝³ : TopologicalSpace R
inst✝² : LinearOrderedField R
inst✝¹ : OrderTopology R
inst✝ : FloorRing R
a : R
ha : 0 ≤ a
⊢ Tendsto (fun x => ↑⌈a * x⌉₊ / x) atTop (𝓝 a)

Tactic:
have A : Tendsto (fun x : R ↦ a + x⁻¹) atTop (𝓝 (a + 0)) :=
  tendsto_const_nhds.add tendsto_inv_atTop_zero

Result:
R : Type u_4
inst✝³ : TopologicalSpace R
inst✝² : LinearOrderedField R
inst✝¹ : OrderTopology R
inst✝ : FloorRing R
a : R
ha : 0 ≤ a
A : Tendsto (fun x => a + x⁻¹) atTop (𝓝 (a + 0))
⊢ Tendsto (fun x => ↑⌈a * x⌉₊ / x) atTop (𝓝 a)

Proof state:
case hgf
R : Type u_4
inst✝³ : TopologicalSpace R
inst✝² : LinearOrderedField R
inst✝¹ : OrderTopology R
inst✝ : FloorRing R
a : R
ha : 0 ≤ a
A : Tendsto (fun x => a + x⁻¹) atTop (𝓝 a)
x : R
hx : x ≥ 1
⊢ a * x ≤ ↑⌈a * x⌉₊

Tactic:
exact Nat.le_ceil _

Result:
no goals

Proof state:
case hfh
R : Type u_4
inst✝³ : TopologicalSpace R
inst✝² : LinearOrderedField R
inst✝¹ : OrderTopology R
inst✝ : FloorRing R
a : R
ha : 0 ≤ a
A : Tendsto (fun x => a + x⁻¹) atTop (𝓝 a)
⊢ ∀ᶠ (b : R) in atTop, ↑⌈a * b⌉₊ / b ≤ a + b⁻¹

Tactic:
refine eventually_atTop.2 ⟨1, fun x hx ↦ ?_⟩

Result:
case hfh
R : Type u_4
inst✝³ : TopologicalSpace R
inst✝² : LinearOrderedField R
inst✝¹ : OrderTopology R
inst✝ : FloorRing R
a : R
ha : 0 ≤ a
A : Tendsto (fun x => a + x⁻¹) atTop (𝓝 a)
x : R
hx : x ≥ 1
⊢ ↑⌈a * x⌉₊ / x ≤ a + x⁻¹

Proof state:
R : Type u_4
inst✝³ : TopologicalSpace R
inst✝² : LinearOrderedField R
inst✝¹ : OrderTopology R
inst✝ : FloorRing R
a : R
ha : 0 ≤ a
A : Tendsto (fun x => a + x⁻¹) atTop (𝓝 (a + 0))
⊢ Tendsto (fun x => ↑⌈a * x⌉₊ / x) atTop (𝓝 a)

Tactic:
rw [add_zero] at A

Result:
R : Type u_4
inst✝³ : TopologicalSpace R
inst✝² : LinearOrderedField R
inst✝¹ : OrderTopology R
inst✝ : FloorRing R
a : R
ha : 0 ≤ a
A : Tendsto (fun x => a + x⁻¹) atTop (𝓝 a)
⊢ Tendsto (fun x => ↑⌈a * x⌉₊ / x) atTop (𝓝 a)

Proof state:
case hgf
R : Type u_4
inst✝³ : TopologicalSpace R
inst✝² : LinearOrderedField R
inst✝¹ : OrderTopology R
inst✝ : FloorRing R
a : R
ha : 0 ≤ a
A : Tendsto (fun x => a + x⁻¹) atTop (𝓝 a)
x : R
hx : x ≥ 1
⊢ a ≤ ↑⌈a * x⌉₊ / x

Tactic:
rw [le_div_iff₀ (zero_lt_one.trans_le hx)]

Result:
case hgf
R : Type u_4
inst✝³ : TopologicalSpace R
inst✝² : LinearOrderedField R
inst✝¹ : OrderTopology R
inst✝ : FloorRing R
a : R
ha : 0 ≤ a
A : Tendsto (fun x => a + x⁻¹) atTop (𝓝 a)
x : R
hx : x ≥ 1
⊢ a * x ≤ ↑⌈a * x⌉₊

Proof state:
case hgf
R : Type u_4
inst✝³ : TopologicalSpace R
inst✝² : LinearOrderedField R
inst✝¹ : OrderTopology R
inst✝ : FloorRing R
a : R
ha : 0 ≤ a
A : Tendsto (fun x => a + x⁻¹) atTop (𝓝 a)
⊢ ∀ᶠ (b : R) in atTop, a ≤ ↑⌈a * b⌉₊ / b

Tactic:
refine eventually_atTop.2 ⟨1, fun x hx ↦ ?_⟩

Result:
case hgf
R : Type u_4
inst✝³ : TopologicalSpace R
inst✝² : LinearOrderedField R
inst✝¹ : OrderTopology R
inst✝ : FloorRing R
a : R
ha : 0 ≤ a
A : Tendsto (fun x => a + x⁻¹) atTop (𝓝 a)
x : R
hx : x ≥ 1
⊢ a ≤ ↑⌈a * x⌉₊ / x

Proof state:
R : Type u_4
inst✝³ : TopologicalSpace R
inst✝² : LinearOrderedField R
inst✝¹ : OrderTopology R
inst✝ : FloorRing R
a : R
ha : 0 ≤ a
A : Tendsto (fun x => a + x⁻¹) atTop (𝓝 a)
⊢ Tendsto (fun x => ↑⌈a * x⌉₊ / x) atTop (𝓝 a)

Tactic:
apply tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds A

Result:
case hgf
R : Type u_4
inst✝³ : TopologicalSpace R
inst✝² : LinearOrderedField R
inst✝¹ : OrderTopology R
inst✝ : FloorRing R
a : R
ha : 0 ≤ a
A : Tendsto (fun x => a + x⁻¹) atTop (𝓝 a)
⊢ ∀ᶠ (b : R) in atTop, a ≤ ↑⌈a * b⌉₊ / b

case hfh
R : Type u_4
inst✝³ : TopologicalSpace R
inst✝² : LinearOrderedField R
inst✝¹ : OrderTopology R
inst✝ : FloorRing R
a : R
ha : 0 ≤ a
A : Tendsto (fun x => a + x⁻¹) atTop (𝓝 a)
⊢ ∀ᶠ (b : R) in atTop, ↑⌈a * b⌉₊ / b ≤ a + b⁻¹

Proof state:
case hfh
R : Type u_4
inst✝³ : TopologicalSpace R
inst✝² : LinearOrderedField R
inst✝¹ : OrderTopology R
inst✝ : FloorRing R
a : R
ha : 0 ≤ a
A : Tendsto (fun x => a + x⁻¹) atTop (𝓝 a)
x : R
hx : x ≥ 1
⊢ ↑⌈a * x⌉₊ / x ≤ a + x⁻¹

Tactic:
simp [div_le_iff₀ (zero_lt_one.trans_le hx), inv_mul_cancel₀ (zero_lt_one.trans_le hx).ne',
  (Nat.ceil_lt_add_one (mul_nonneg ha (zero_le_one.trans hx))).le, add_mul]

Result:
no goals