l3lab/ntp-mathlib huggingface dataset:
theorem exists_isLUB (S : Set ℝ) (hne : S.Nonempty) (hbdd : BddAbove S) : ∃ x, IsLUB S x := by
  rcases hne, hbdd with ⟨⟨L, hL⟩, ⟨U, hU⟩⟩
  have : ∀ d : ℕ, BddAbove { m : ℤ | ∃ y ∈ S, (m : ℝ) ≤ y * d } := by
    cases' exists_int_gt U with k hk
    refine' fun d => ⟨k * d, fun z h => _⟩
    rcases h with ⟨y, yS, hy⟩
    refine' Int.cast_le.1 (hy.trans _)
    push_cast
    exact mul_le_mul_of_nonneg_right ((hU yS).trans hk.le) d.cast_nonneg
  choose f hf using fun d : ℕ =>
    Int.exists_greatest_of_bdd (this d) ⟨⌊L * d⌋, L, hL, Int.floor_le _⟩
  have hf₁ : ∀ n > 0, ∃ y ∈ S, ((f n / n : ℚ) : ℝ) ≤ y := fun n n0 =>
    let ⟨y, yS, hy⟩ := (hf n).1
    ⟨y, yS, by simpa using (div_le_iff (Nat.cast_pos.2 n0 : (_ : ℝ) < _)).2 hy⟩
  have hf₂ : ∀ n > 0, ∀ y ∈ S, (y - ((n : ℕ) : ℝ)⁻¹) < (f n / n : ℚ) := by
    intro n n0 y yS
    have := (Int.sub_one_lt_floor _).trans_le (Int.cast_le.2 <| (hf n).2 _ ⟨y, yS, Int.floor_le _⟩)
    simp only [Rat.cast_div, Rat.cast_coe_int, Rat.cast_coe_nat, gt_iff_lt]
    rwa [lt_div_iff (Nat.cast_pos.2 n0 : (_ : ℝ) < _), sub_mul, _root_.inv_mul_cancel]
    exact ne_of_gt (Nat.cast_pos.2 n0)
  have hg : IsCauSeq abs (fun n => f n / n : ℕ → ℚ) := by
    intro ε ε0
    suffices ∀ j ≥ ⌈ε⁻¹⌉₊, ∀ k ≥ ⌈ε⁻¹⌉₊, (f j / j - f k / k : ℚ) < ε by
      refine' ⟨_, fun j ij => abs_lt.2 ⟨_, this _ ij _ le_rfl⟩⟩
      rw [neg_lt, neg_sub]
      exact this _ le_rfl _ ij
    intro j ij k ik
    replace ij := le_trans (Nat.le_ceil _) (Nat.cast_le.2 ij)
    replace ik := le_trans (Nat.le_ceil _) (Nat.cast_le.2 ik)
    have j0 := Nat.cast_pos.1 ((inv_pos.2 ε0).trans_le ij)
    have k0 := Nat.cast_pos.1 ((inv_pos.2 ε0).trans_le ik)
    rcases hf₁ _ j0 with ⟨y, yS, hy⟩
    refine' lt_of_lt_of_le ((@Rat.cast_lt ℝ _ _ _).1 _) ((inv_le ε0 (Nat.cast_pos.2 k0)).1 ik)
    simpa using sub_lt_iff_lt_add'.2 (lt_of_le_of_lt hy <| sub_lt_iff_lt_add.1 <| hf₂ _ k0 _ yS)
  let g : CauSeq ℚ abs := ⟨fun n => f n / n, hg⟩
  

Proof state:
case intro.intro
S : Set ℝ
L : ℝ
hL : L ∈ S
U : ℝ
hU : U ∈ upperBounds S
this : ∀ (d : ℕ), BddAbove {m | ∃ y ∈ S, ↑m ≤ y * ↑d}
f : ℕ → ℤ
hf : ∀ (d : ℕ), f d ∈ {m | ∃ y ∈ S, ↑m ≤ y * ↑d} ∧ ∀ z ∈ {m | ∃ y ∈ S, ↑m ≤ y * ↑d}, z ≤ f d
hf₁ : ∀ n > 0, ∃ y ∈ S, ↑(↑(f n) / ↑n) ≤ y
hf₂ : ∀ n > 0, ∀ y ∈ S, y - (↑n)⁻¹ < ↑(↑(f n) / ↑n)
hg : IsCauSeq abs fun n => ↑(f n) / ↑n
g : CauSeq ℚ abs := { val := fun n => ↑(f n) / ↑n, property := hg }
⊢ ∃ x, IsLUB S x

Next tactic:
refine' ⟨mk g, ⟨fun x xS => _, fun y h => _⟩⟩
theorem exists_isLUB (S : Set ℝ) (hne : S.Nonempty) (hbdd : BddAbove S) : ∃ x, IsLUB S x := by
  rcases hne, hbdd with ⟨⟨L, hL⟩, ⟨U, hU⟩⟩
  have : ∀ d : ℕ, BddAbove { m : ℤ | ∃ y ∈ S, (m : ℝ) ≤ y * d } := by
    cases' exists_int_gt U with k hk
    refine' fun d => ⟨k * d, fun z h => _⟩
    rcases h with ⟨y, yS, hy⟩
    refine' Int.cast_le.1 (hy.trans _)
    push_cast
    exact mul_le_mul_of_nonneg_right ((hU yS).trans hk.le) d.cast_nonneg
  choose f hf using fun d : ℕ =>
    Int.exists_greatest_of_bdd (this d) ⟨⌊L * d⌋, L, hL, Int.floor_le _⟩
  have hf₁ : ∀ n > 0, ∃ y ∈ S, ((f n / n : ℚ) : ℝ) ≤ y := fun n n0 =>
    let ⟨y, yS, hy⟩ := (hf n).1
    ⟨y, yS, by simpa using (div_le_iff (Nat.cast_pos.2 n0 : (_ : ℝ) < _)).2 hy⟩
  have hf₂ : ∀ n > 0, ∀ y ∈ S, (y - ((n : ℕ) : ℝ)⁻¹) < (f n / n : ℚ) := by
    intro n n0 y yS
    have := (Int.sub_one_lt_floor _).trans_le (Int.cast_le.2 <| (hf n).2 _ ⟨y, yS, Int.floor_le _⟩)
    simp only [Rat.cast_div, Rat.cast_coe_int, Rat.cast_coe_nat, gt_iff_lt]
    rwa [lt_div_iff (Nat.cast_pos.2 n0 : (_ : ℝ) < _), sub_mul, _root_.inv_mul_cancel]
    exact ne_of_gt (Nat.cast_pos.2 n0)
  have hg : IsCauSeq abs (fun n => f n / n : ℕ → ℚ) := by
    intro ε ε0
    suffices ∀ j ≥ ⌈ε⁻¹⌉₊, ∀ k ≥ ⌈ε⁻¹⌉₊, (f j / j - f k / k : ℚ) < ε by
      refine' ⟨_, fun j ij => abs_lt.2 ⟨_, this _ ij _ le_rfl⟩⟩
      rw [neg_lt, neg_sub]
      exact this _ le_rfl _ ij
    intro j ij k ik
    replace ij := le_trans (Nat.le_ceil _) (Nat.cast_le.2 ij)
    replace ik := le_trans (Nat.le_ceil _) (Nat.cast_le.2 ik)
    have j0 := Nat.cast_pos.1 ((inv_pos.2 ε0).trans_le ij)
    have k0 := Nat.cast_pos.1 ((inv_pos.2 ε0).trans_le ik)
    rcases hf₁ _ j0 with ⟨y, yS, hy⟩
    refine' lt_of_lt_of_le ((@Rat.cast_lt ℝ _ _ _).1 _) ((inv_le ε0 (Nat.cast_pos.2 k0)).1 ik)
    simpa using sub_lt_iff_lt_add'.2 (lt_of_le_of_lt hy <| sub_lt_iff_lt_add.1 <| hf₂ _ k0 _ yS)
  let g : CauSeq ℚ abs := ⟨fun n => f n / n, hg⟩
  refine' ⟨mk g, ⟨fun x xS => _, fun y h => _⟩⟩
  · 

Proof state:
case intro.intro.refine'_1
S : Set ℝ
L : ℝ
hL : L ∈ S
U : ℝ
hU : U ∈ upperBounds S
this : ∀ (d : ℕ), BddAbove {m | ∃ y ∈ S, ↑m ≤ y * ↑d}
f : ℕ → ℤ
hf : ∀ (d : ℕ), f d ∈ {m | ∃ y ∈ S, ↑m ≤ y * ↑d} ∧ ∀ z ∈ {m | ∃ y ∈ S, ↑m ≤ y * ↑d}, z ≤ f d
hf₁ : ∀ n > 0, ∃ y ∈ S, ↑(↑(f n) / ↑n) ≤ y
hf₂ : ∀ n > 0, ∀ y ∈ S, y - (↑n)⁻¹ < ↑(↑(f n) / ↑n)
hg : IsCauSeq abs fun n => ↑(f n) / ↑n
g : CauSeq ℚ abs := { val := fun n => ↑(f n) / ↑n, property := hg }
x : ℝ
xS : x ∈ S
⊢ x ≤ mk g

Next tactic:
refine' le_of_forall_ge_of_dense fun z xz => _
theorem exists_isLUB (S : Set ℝ) (hne : S.Nonempty) (hbdd : BddAbove S) : ∃ x, IsLUB S x := by
  rcases hne, hbdd with ⟨⟨L, hL⟩, ⟨U, hU⟩⟩
  have : ∀ d : ℕ, BddAbove { m : ℤ | ∃ y ∈ S, (m : ℝ) ≤ y * d } := by
    cases' exists_int_gt U with k hk
    refine' fun d => ⟨k * d, fun z h => _⟩
    rcases h with ⟨y, yS, hy⟩
    refine' Int.cast_le.1 (hy.trans _)
    push_cast
    exact mul_le_mul_of_nonneg_right ((hU yS).trans hk.le) d.cast_nonneg
  choose f hf using fun d : ℕ =>
    Int.exists_greatest_of_bdd (this d) ⟨⌊L * d⌋, L, hL, Int.floor_le _⟩
  have hf₁ : ∀ n > 0, ∃ y ∈ S, ((f n / n : ℚ) : ℝ) ≤ y := fun n n0 =>
    let ⟨y, yS, hy⟩ := (hf n).1
    ⟨y, yS, by simpa using (div_le_iff (Nat.cast_pos.2 n0 : (_ : ℝ) < _)).2 hy⟩
  have hf₂ : ∀ n > 0, ∀ y ∈ S, (y - ((n : ℕ) : ℝ)⁻¹) < (f n / n : ℚ) := by
    intro n n0 y yS
    have := (Int.sub_one_lt_floor _).trans_le (Int.cast_le.2 <| (hf n).2 _ ⟨y, yS, Int.floor_le _⟩)
    simp only [Rat.cast_div, Rat.cast_coe_int, Rat.cast_coe_nat, gt_iff_lt]
    rwa [lt_div_iff (Nat.cast_pos.2 n0 : (_ : ℝ) < _), sub_mul, _root_.inv_mul_cancel]
    exact ne_of_gt (Nat.cast_pos.2 n0)
  have hg : IsCauSeq abs (fun n => f n / n : ℕ → ℚ) := by
    intro ε ε0
    suffices ∀ j ≥ ⌈ε⁻¹⌉₊, ∀ k ≥ ⌈ε⁻¹⌉₊, (f j / j - f k / k : ℚ) < ε by
      refine' ⟨_, fun j ij => abs_lt.2 ⟨_, this _ ij _ le_rfl⟩⟩
      rw [neg_lt, neg_sub]
      exact this _ le_rfl _ ij
    intro j ij k ik
    replace ij := le_trans (Nat.le_ceil _) (Nat.cast_le.2 ij)
    replace ik := le_trans (Nat.le_ceil _) (Nat.cast_le.2 ik)
    have j0 := Nat.cast_pos.1 ((inv_pos.2 ε0).trans_le ij)
    have k0 := Nat.cast_pos.1 ((inv_pos.2 ε0).trans_le ik)
    rcases hf₁ _ j0 with ⟨y, yS, hy⟩
    refine' lt_of_lt_of_le ((@Rat.cast_lt ℝ _ _ _).1 _) ((inv_le ε0 (Nat.cast_pos.2 k0)).1 ik)
    simpa using sub_lt_iff_lt_add'.2 (lt_of_le_of_lt hy <| sub_lt_iff_lt_add.1 <| hf₂ _ k0 _ yS)
  let g : CauSeq ℚ abs := ⟨fun n => f n / n, hg⟩
  refine' ⟨mk g, ⟨fun x xS => _, fun y h => _⟩⟩
  · refine' le_of_forall_ge_of_dense fun z xz => _
    

Proof state:
case intro.intro.refine'_1
S : Set ℝ
L : ℝ
hL : L ∈ S
U : ℝ
hU : U ∈ upperBounds S
this : ∀ (d : ℕ), BddAbove {m | ∃ y ∈ S, ↑m ≤ y * ↑d}
f : ℕ → ℤ
hf : ∀ (d : ℕ), f d ∈ {m | ∃ y ∈ S, ↑m ≤ y * ↑d} ∧ ∀ z ∈ {m | ∃ y ∈ S, ↑m ≤ y * ↑d}, z ≤ f d
hf₁ : ∀ n > 0, ∃ y ∈ S, ↑(↑(f n) / ↑n) ≤ y
hf₂ : ∀ n > 0, ∀ y ∈ S, y - (↑n)⁻¹ < ↑(↑(f n) / ↑n)
hg : IsCauSeq abs fun n => ↑(f n) / ↑n
g : CauSeq ℚ abs := { val := fun n => ↑(f n) / ↑n, property := hg }
x : ℝ
xS : x ∈ S
z : ℝ
xz : z < x
⊢ z ≤ mk g

Next tactic:
cases' exists_nat_gt (x - z)⁻¹ with K hK
theorem exists_isLUB (S : Set ℝ) (hne : S.Nonempty) (hbdd : BddAbove S) : ∃ x, IsLUB S x := by
  rcases hne, hbdd with ⟨⟨L, hL⟩, ⟨U, hU⟩⟩
  have : ∀ d : ℕ, BddAbove { m : ℤ | ∃ y ∈ S, (m : ℝ) ≤ y * d } := by
    cases' exists_int_gt U with k hk
    refine' fun d => ⟨k * d, fun z h => _⟩
    rcases h with ⟨y, yS, hy⟩
    refine' Int.cast_le.1 (hy.trans _)
    push_cast
    exact mul_le_mul_of_nonneg_right ((hU yS).trans hk.le) d.cast_nonneg
  choose f hf using fun d : ℕ =>
    Int.exists_greatest_of_bdd (this d) ⟨⌊L * d⌋, L, hL, Int.floor_le _⟩
  have hf₁ : ∀ n > 0, ∃ y ∈ S, ((f n / n : ℚ) : ℝ) ≤ y := fun n n0 =>
    let ⟨y, yS, hy⟩ := (hf n).1
    ⟨y, yS, by simpa using (div_le_iff (Nat.cast_pos.2 n0 : (_ : ℝ) < _)).2 hy⟩
  have hf₂ : ∀ n > 0, ∀ y ∈ S, (y - ((n : ℕ) : ℝ)⁻¹) < (f n / n : ℚ) := by
    intro n n0 y yS
    have := (Int.sub_one_lt_floor _).trans_le (Int.cast_le.2 <| (hf n).2 _ ⟨y, yS, Int.floor_le _⟩)
    simp only [Rat.cast_div, Rat.cast_coe_int, Rat.cast_coe_nat, gt_iff_lt]
    rwa [lt_div_iff (Nat.cast_pos.2 n0 : (_ : ℝ) < _), sub_mul, _root_.inv_mul_cancel]
    exact ne_of_gt (Nat.cast_pos.2 n0)
  have hg : IsCauSeq abs (fun n => f n / n : ℕ → ℚ) := by
    intro ε ε0
    suffices ∀ j ≥ ⌈ε⁻¹⌉₊, ∀ k ≥ ⌈ε⁻¹⌉₊, (f j / j - f k / k : ℚ) < ε by
      refine' ⟨_, fun j ij => abs_lt.2 ⟨_, this _ ij _ le_rfl⟩⟩
      rw [neg_lt, neg_sub]
      exact this _ le_rfl _ ij
    intro j ij k ik
    replace ij := le_trans (Nat.le_ceil _) (Nat.cast_le.2 ij)
    replace ik := le_trans (Nat.le_ceil _) (Nat.cast_le.2 ik)
    have j0 := Nat.cast_pos.1 ((inv_pos.2 ε0).trans_le ij)
    have k0 := Nat.cast_pos.1 ((inv_pos.2 ε0).trans_le ik)
    rcases hf₁ _ j0 with ⟨y, yS, hy⟩
    refine' lt_of_lt_of_le ((@Rat.cast_lt ℝ _ _ _).1 _) ((inv_le ε0 (Nat.cast_pos.2 k0)).1 ik)
    simpa using sub_lt_iff_lt_add'.2 (lt_of_le_of_lt hy <| sub_lt_iff_lt_add.1 <| hf₂ _ k0 _ yS)
  let g : CauSeq ℚ abs := ⟨fun n => f n / n, hg⟩
  refine' ⟨mk g, ⟨fun x xS => _, fun y h => _⟩⟩
  · refine' le_of_forall_ge_of_dense fun z xz => _
    cases' exists_nat_gt (x - z)⁻¹ with K hK
    

Proof state:
case intro.intro.refine'_1.intro
S : Set ℝ
L : ℝ
hL : L ∈ S
U : ℝ
hU : U ∈ upperBounds S
this : ∀ (d : ℕ), BddAbove {m | ∃ y ∈ S, ↑m ≤ y * ↑d}
f : ℕ → ℤ
hf : ∀ (d : ℕ), f d ∈ {m | ∃ y ∈ S, ↑m ≤ y * ↑d} ∧ ∀ z ∈ {m | ∃ y ∈ S, ↑m ≤ y * ↑d}, z ≤ f d
hf₁ : ∀ n > 0, ∃ y ∈ S, ↑(↑(f n) / ↑n) ≤ y
hf₂ : ∀ n > 0, ∀ y ∈ S, y - (↑n)⁻¹ < ↑(↑(f n) / ↑n)
hg : IsCauSeq abs fun n => ↑(f n) / ↑n
g : CauSeq ℚ abs := { val := fun n => ↑(f n) / ↑n, property := hg }
x : ℝ
xS : x ∈ S
z : ℝ
xz : z < x
K : ℕ
hK : (x - z)⁻¹ < ↑K
⊢ z ≤ mk g

Next tactic:
refine' le_mk_of_forall_le ⟨K, fun n nK => _⟩
theorem exists_isLUB (S : Set ℝ) (hne : S.Nonempty) (hbdd : BddAbove S) : ∃ x, IsLUB S x := by
  rcases hne, hbdd with ⟨⟨L, hL⟩, ⟨U, hU⟩⟩
  have : ∀ d : ℕ, BddAbove { m : ℤ | ∃ y ∈ S, (m : ℝ) ≤ y * d } := by
    cases' exists_int_gt U with k hk
    refine' fun d => ⟨k * d, fun z h => _⟩
    rcases h with ⟨y, yS, hy⟩
    refine' Int.cast_le.1 (hy.trans _)
    push_cast
    exact mul_le_mul_of_nonneg_right ((hU yS).trans hk.le) d.cast_nonneg
  choose f hf using fun d : ℕ =>
    Int.exists_greatest_of_bdd (this d) ⟨⌊L * d⌋, L, hL, Int.floor_le _⟩
  have hf₁ : ∀ n > 0, ∃ y ∈ S, ((f n / n : ℚ) : ℝ) ≤ y := fun n n0 =>
    let ⟨y, yS, hy⟩ := (hf n).1
    ⟨y, yS, by simpa using (div_le_iff (Nat.cast_pos.2 n0 : (_ : ℝ) < _)).2 hy⟩
  have hf₂ : ∀ n > 0, ∀ y ∈ S, (y - ((n : ℕ) : ℝ)⁻¹) < (f n / n : ℚ) := by
    intro n n0 y yS
    have := (Int.sub_one_lt_floor _).trans_le (Int.cast_le.2 <| (hf n).2 _ ⟨y, yS, Int.floor_le _⟩)
    simp only [Rat.cast_div, Rat.cast_coe_int, Rat.cast_coe_nat, gt_iff_lt]
    rwa [lt_div_iff (Nat.cast_pos.2 n0 : (_ : ℝ) < _), sub_mul, _root_.inv_mul_cancel]
    exact ne_of_gt (Nat.cast_pos.2 n0)
  have hg : IsCauSeq abs (fun n => f n / n : ℕ → ℚ) := by
    intro ε ε0
    suffices ∀ j ≥ ⌈ε⁻¹⌉₊, ∀ k ≥ ⌈ε⁻¹⌉₊, (f j / j - f k / k : ℚ) < ε by
      refine' ⟨_, fun j ij => abs_lt.2 ⟨_, this _ ij _ le_rfl⟩⟩
      rw [neg_lt, neg_sub]
      exact this _ le_rfl _ ij
    intro j ij k ik
    replace ij := le_trans (Nat.le_ceil _) (Nat.cast_le.2 ij)
    replace ik := le_trans (Nat.le_ceil _) (Nat.cast_le.2 ik)
    have j0 := Nat.cast_pos.1 ((inv_pos.2 ε0).trans_le ij)
    have k0 := Nat.cast_pos.1 ((inv_pos.2 ε0).trans_le ik)
    rcases hf₁ _ j0 with ⟨y, yS, hy⟩
    refine' lt_of_lt_of_le ((@Rat.cast_lt ℝ _ _ _).1 _) ((inv_le ε0 (Nat.cast_pos.2 k0)).1 ik)
    simpa using sub_lt_iff_lt_add'.2 (lt_of_le_of_lt hy <| sub_lt_iff_lt_add.1 <| hf₂ _ k0 _ yS)
  let g : CauSeq ℚ abs := ⟨fun n => f n / n, hg⟩
  refine' ⟨mk g, ⟨fun x xS => _, fun y h => _⟩⟩
  · refine' le_of_forall_ge_of_dense fun z xz => _
    cases' exists_nat_gt (x - z)⁻¹ with K hK
    refine' le_mk_of_forall_le ⟨K, fun n nK => _⟩
    

Proof state:
case intro.intro.refine'_1.intro
S : Set ℝ
L : ℝ
hL : L ∈ S
U : ℝ
hU : U ∈ upperBounds S
this : ∀ (d : ℕ), BddAbove {m | ∃ y ∈ S, ↑m ≤ y * ↑d}
f : ℕ → ℤ
hf : ∀ (d : ℕ), f d ∈ {m | ∃ y ∈ S, ↑m ≤ y * ↑d} ∧ ∀ z ∈ {m | ∃ y ∈ S, ↑m ≤ y * ↑d}, z ≤ f d
hf₁ : ∀ n > 0, ∃ y ∈ S, ↑(↑(f n) / ↑n) ≤ y
hf₂ : ∀ n > 0, ∀ y ∈ S, y - (↑n)⁻¹ < ↑(↑(f n) / ↑n)
hg : IsCauSeq abs fun n => ↑(f n) / ↑n
g : CauSeq ℚ abs := { val := fun n => ↑(f n) / ↑n, property := hg }
x : ℝ
xS : x ∈ S
z : ℝ
xz : z < x
K : ℕ
hK : (x - z)⁻¹ < ↑K
n : ℕ
nK : n ≥ K
⊢ z ≤ ↑(↑g n)

Next tactic:
replace xz := sub_pos.2 xz
theorem exists_isLUB (S : Set ℝ) (hne : S.Nonempty) (hbdd : BddAbove S) : ∃ x, IsLUB S x := by
  rcases hne, hbdd with ⟨⟨L, hL⟩, ⟨U, hU⟩⟩
  have : ∀ d : ℕ, BddAbove { m : ℤ | ∃ y ∈ S, (m : ℝ) ≤ y * d } := by
    cases' exists_int_gt U with k hk
    refine' fun d => ⟨k * d, fun z h => _⟩
    rcases h with ⟨y, yS, hy⟩
    refine' Int.cast_le.1 (hy.trans _)
    push_cast
    exact mul_le_mul_of_nonneg_right ((hU yS).trans hk.le) d.cast_nonneg
  choose f hf using fun d : ℕ =>
    Int.exists_greatest_of_bdd (this d) ⟨⌊L * d⌋, L, hL, Int.floor_le _⟩
  have hf₁ : ∀ n > 0, ∃ y ∈ S, ((f n / n : ℚ) : ℝ) ≤ y := fun n n0 =>
    let ⟨y, yS, hy⟩ := (hf n).1
    ⟨y, yS, by simpa using (div_le_iff (Nat.cast_pos.2 n0 : (_ : ℝ) < _)).2 hy⟩
  have hf₂ : ∀ n > 0, ∀ y ∈ S, (y - ((n : ℕ) : ℝ)⁻¹) < (f n / n : ℚ) := by
    intro n n0 y yS
    have := (Int.sub_one_lt_floor _).trans_le (Int.cast_le.2 <| (hf n).2 _ ⟨y, yS, Int.floor_le _⟩)
    simp only [Rat.cast_div, Rat.cast_coe_int, Rat.cast_coe_nat, gt_iff_lt]
    rwa [lt_div_iff (Nat.cast_pos.2 n0 : (_ : ℝ) < _), sub_mul, _root_.inv_mul_cancel]
    exact ne_of_gt (Nat.cast_pos.2 n0)
  have hg : IsCauSeq abs (fun n => f n / n : ℕ → ℚ) := by
    intro ε ε0
    suffices ∀ j ≥ ⌈ε⁻¹⌉₊, ∀ k ≥ ⌈ε⁻¹⌉₊, (f j / j - f k / k : ℚ) < ε by
      refine' ⟨_, fun j ij => abs_lt.2 ⟨_, this _ ij _ le_rfl⟩⟩
      rw [neg_lt, neg_sub]
      exact this _ le_rfl _ ij
    intro j ij k ik
    replace ij := le_trans (Nat.le_ceil _) (Nat.cast_le.2 ij)
    replace ik := le_trans (Nat.le_ceil _) (Nat.cast_le.2 ik)
    have j0 := Nat.cast_pos.1 ((inv_pos.2 ε0).trans_le ij)
    have k0 := Nat.cast_pos.1 ((inv_pos.2 ε0).trans_le ik)
    rcases hf₁ _ j0 with ⟨y, yS, hy⟩
    refine' lt_of_lt_of_le ((@Rat.cast_lt ℝ _ _ _).1 _) ((inv_le ε0 (Nat.cast_pos.2 k0)).1 ik)
    simpa using sub_lt_iff_lt_add'.2 (lt_of_le_of_lt hy <| sub_lt_iff_lt_add.1 <| hf₂ _ k0 _ yS)
  let g : CauSeq ℚ abs := ⟨fun n => f n / n, hg⟩
  refine' ⟨mk g, ⟨fun x xS => _, fun y h => _⟩⟩
  · refine' le_of_forall_ge_of_dense fun z xz => _
    cases' exists_nat_gt (x - z)⁻¹ with K hK
    refine' le_mk_of_forall_le ⟨K, fun n nK => _⟩
    replace xz := sub_pos.2 xz
    

Proof state:
case intro.intro.refine'_1.intro
S : Set ℝ
L : ℝ
hL : L ∈ S
U : ℝ
hU : U ∈ upperBounds S
this : ∀ (d : ℕ), BddAbove {m | ∃ y ∈ S, ↑m ≤ y * ↑d}
f : ℕ → ℤ
hf : ∀ (d : ℕ), f d ∈ {m | ∃ y ∈ S, ↑m ≤ y * ↑d} ∧ ∀ z ∈ {m | ∃ y ∈ S, ↑m ≤ y * ↑d}, z ≤ f d
hf₁ : ∀ n > 0, ∃ y ∈ S, ↑(↑(f n) / ↑n) ≤ y
hf₂ : ∀ n > 0, ∀ y ∈ S, y - (↑n)⁻¹ < ↑(↑(f n) / ↑n)
hg : IsCauSeq abs fun n => ↑(f n) / ↑n
g : CauSeq ℚ abs := { val := fun n => ↑(f n) / ↑n, property := hg }
x : ℝ
xS : x ∈ S
z : ℝ
K : ℕ
hK : (x - z)⁻¹ < ↑K
n : ℕ
nK : n ≥ K
xz : 0 < x - z
⊢ z ≤ ↑(↑g n)

Next tactic:
replace hK := hK.le.trans (Nat.cast_le.2 nK)
theorem exists_isLUB (S : Set ℝ) (hne : S.Nonempty) (hbdd : BddAbove S) : ∃ x, IsLUB S x := by
  rcases hne, hbdd with ⟨⟨L, hL⟩, ⟨U, hU⟩⟩
  have : ∀ d : ℕ, BddAbove { m : ℤ | ∃ y ∈ S, (m : ℝ) ≤ y * d } := by
    cases' exists_int_gt U with k hk
    refine' fun d => ⟨k * d, fun z h => _⟩
    rcases h with ⟨y, yS, hy⟩
    refine' Int.cast_le.1 (hy.trans _)
    push_cast
    exact mul_le_mul_of_nonneg_right ((hU yS).trans hk.le) d.cast_nonneg
  choose f hf using fun d : ℕ =>
    Int.exists_greatest_of_bdd (this d) ⟨⌊L * d⌋, L, hL, Int.floor_le _⟩
  have hf₁ : ∀ n > 0, ∃ y ∈ S, ((f n / n : ℚ) : ℝ) ≤ y := fun n n0 =>
    let ⟨y, yS, hy⟩ := (hf n).1
    ⟨y, yS, by simpa using (div_le_iff (Nat.cast_pos.2 n0 : (_ : ℝ) < _)).2 hy⟩
  have hf₂ : ∀ n > 0, ∀ y ∈ S, (y - ((n : ℕ) : ℝ)⁻¹) < (f n / n : ℚ) := by
    intro n n0 y yS
    have := (Int.sub_one_lt_floor _).trans_le (Int.cast_le.2 <| (hf n).2 _ ⟨y, yS, Int.floor_le _⟩)
    simp only [Rat.cast_div, Rat.cast_coe_int, Rat.cast_coe_nat, gt_iff_lt]
    rwa [lt_div_iff (Nat.cast_pos.2 n0 : (_ : ℝ) < _), sub_mul, _root_.inv_mul_cancel]
    exact ne_of_gt (Nat.cast_pos.2 n0)
  have hg : IsCauSeq abs (fun n => f n / n : ℕ → ℚ) := by
    intro ε ε0
    suffices ∀ j ≥ ⌈ε⁻¹⌉₊, ∀ k ≥ ⌈ε⁻¹⌉₊, (f j / j - f k / k : ℚ) < ε by
      refine' ⟨_, fun j ij => abs_lt.2 ⟨_, this _ ij _ le_rfl⟩⟩
      rw [neg_lt, neg_sub]
      exact this _ le_rfl _ ij
    intro j ij k ik
    replace ij := le_trans (Nat.le_ceil _) (Nat.cast_le.2 ij)
    replace ik := le_trans (Nat.le_ceil _) (Nat.cast_le.2 ik)
    have j0 := Nat.cast_pos.1 ((inv_pos.2 ε0).trans_le ij)
    have k0 := Nat.cast_pos.1 ((inv_pos.2 ε0).trans_le ik)
    rcases hf₁ _ j0 with ⟨y, yS, hy⟩
    refine' lt_of_lt_of_le ((@Rat.cast_lt ℝ _ _ _).1 _) ((inv_le ε0 (Nat.cast_pos.2 k0)).1 ik)
    simpa using sub_lt_iff_lt_add'.2 (lt_of_le_of_lt hy <| sub_lt_iff_lt_add.1 <| hf₂ _ k0 _ yS)
  let g : CauSeq ℚ abs := ⟨fun n => f n / n, hg⟩
  refine' ⟨mk g, ⟨fun x xS => _, fun y h => _⟩⟩
  · refine' le_of_forall_ge_of_dense fun z xz => _
    cases' exists_nat_gt (x - z)⁻¹ with K hK
    refine' le_mk_of_forall_le ⟨K, fun n nK => _⟩
    replace xz := sub_pos.2 xz
    replace hK := hK.le.trans (Nat.cast_le.2 nK)
    

Proof state:
case intro.intro.refine'_1.intro
S : Set ℝ
L : ℝ
hL : L ∈ S
U : ℝ
hU : U ∈ upperBounds S
this : ∀ (d : ℕ), BddAbove {m | ∃ y ∈ S, ↑m ≤ y * ↑d}
f : ℕ → ℤ
hf : ∀ (d : ℕ), f d ∈ {m | ∃ y ∈ S, ↑m ≤ y * ↑d} ∧ ∀ z ∈ {m | ∃ y ∈ S, ↑m ≤ y * ↑d}, z ≤ f d
hf₁ : ∀ n > 0, ∃ y ∈ S, ↑(↑(f n) / ↑n) ≤ y
hf₂ : ∀ n > 0, ∀ y ∈ S, y - (↑n)⁻¹ < ↑(↑(f n) / ↑n)
hg : IsCauSeq abs fun n => ↑(f n) / ↑n
g : CauSeq ℚ abs := { val := fun n => ↑(f n) / ↑n, property := hg }
x : ℝ
xS : x ∈ S
z : ℝ
K n : ℕ
nK : n ≥ K
xz : 0 < x - z
hK : (x - z)⁻¹ ≤ ↑n
⊢ z ≤ ↑(↑g n)

Next tactic:
have n0 : 0 < n := Nat.cast_pos.1 ((inv_pos.2 xz).trans_le hK)
theorem exists_isLUB (S : Set ℝ) (hne : S.Nonempty) (hbdd : BddAbove S) : ∃ x, IsLUB S x := by
  rcases hne, hbdd with ⟨⟨L, hL⟩, ⟨U, hU⟩⟩
  have : ∀ d : ℕ, BddAbove { m : ℤ | ∃ y ∈ S, (m : ℝ) ≤ y * d } := by
    cases' exists_int_gt U with k hk
    refine' fun d => ⟨k * d, fun z h => _⟩
    rcases h with ⟨y, yS, hy⟩
    refine' Int.cast_le.1 (hy.trans _)
    push_cast
    exact mul_le_mul_of_nonneg_right ((hU yS).trans hk.le) d.cast_nonneg
  choose f hf using fun d : ℕ =>
    Int.exists_greatest_of_bdd (this d) ⟨⌊L * d⌋, L, hL, Int.floor_le _⟩
  have hf₁ : ∀ n > 0, ∃ y ∈ S, ((f n / n : ℚ) : ℝ) ≤ y := fun n n0 =>
    let ⟨y, yS, hy⟩ := (hf n).1
    ⟨y, yS, by simpa using (div_le_iff (Nat.cast_pos.2 n0 : (_ : ℝ) < _)).2 hy⟩
  have hf₂ : ∀ n > 0, ∀ y ∈ S, (y - ((n : ℕ) : ℝ)⁻¹) < (f n / n : ℚ) := by
    intro n n0 y yS
    have := (Int.sub_one_lt_floor _).trans_le (Int.cast_le.2 <| (hf n).2 _ ⟨y, yS, Int.floor_le _⟩)
    simp only [Rat.cast_div, Rat.cast_coe_int, Rat.cast_coe_nat, gt_iff_lt]
    rwa [lt_div_iff (Nat.cast_pos.2 n0 : (_ : ℝ) < _), sub_mul, _root_.inv_mul_cancel]
    exact ne_of_gt (Nat.cast_pos.2 n0)
  have hg : IsCauSeq abs (fun n => f n / n : ℕ → ℚ) := by
    intro ε ε0
    suffices ∀ j ≥ ⌈ε⁻¹⌉₊, ∀ k ≥ ⌈ε⁻¹⌉₊, (f j / j - f k / k : ℚ) < ε by
      refine' ⟨_, fun j ij => abs_lt.2 ⟨_, this _ ij _ le_rfl⟩⟩
      rw [neg_lt, neg_sub]
      exact this _ le_rfl _ ij
    intro j ij k ik
    replace ij := le_trans (Nat.le_ceil _) (Nat.cast_le.2 ij)
    replace ik := le_trans (Nat.le_ceil _) (Nat.cast_le.2 ik)
    have j0 := Nat.cast_pos.1 ((inv_pos.2 ε0).trans_le ij)
    have k0 := Nat.cast_pos.1 ((inv_pos.2 ε0).trans_le ik)
    rcases hf₁ _ j0 with ⟨y, yS, hy⟩
    refine' lt_of_lt_of_le ((@Rat.cast_lt ℝ _ _ _).1 _) ((inv_le ε0 (Nat.cast_pos.2 k0)).1 ik)
    simpa using sub_lt_iff_lt_add'.2 (lt_of_le_of_lt hy <| sub_lt_iff_lt_add.1 <| hf₂ _ k0 _ yS)
  let g : CauSeq ℚ abs := ⟨fun n => f n / n, hg⟩
  refine' ⟨mk g, ⟨fun x xS => _, fun y h => _⟩⟩
  · refine' le_of_forall_ge_of_dense fun z xz => _
    cases' exists_nat_gt (x - z)⁻¹ with K hK
    refine' le_mk_of_forall_le ⟨K, fun n nK => _⟩
    replace xz := sub_pos.2 xz
    replace hK := hK.le.trans (Nat.cast_le.2 nK)
    have n0 : 0 < n := Nat.cast_pos.1 ((inv_pos.2 xz).trans_le hK)
    

Proof state:
case intro.intro.refine'_1.intro
S : Set ℝ
L : ℝ
hL : L ∈ S
U : ℝ
hU : U ∈ upperBounds S
this : ∀ (d : ℕ), BddAbove {m | ∃ y ∈ S, ↑m ≤ y * ↑d}
f : ℕ → ℤ
hf : ∀ (d : ℕ), f d ∈ {m | ∃ y ∈ S, ↑m ≤ y * ↑d} ∧ ∀ z ∈ {m | ∃ y ∈ S, ↑m ≤ y * ↑d}, z ≤ f d
hf₁ : ∀ n > 0, ∃ y ∈ S, ↑(↑(f n) / ↑n) ≤ y
hf₂ : ∀ n > 0, ∀ y ∈ S, y - (↑n)⁻¹ < ↑(↑(f n) / ↑n)
hg : IsCauSeq abs fun n => ↑(f n) / ↑n
g : CauSeq ℚ abs := { val := fun n => ↑(f n) / ↑n, property := hg }
x : ℝ
xS : x ∈ S
z : ℝ
K n : ℕ
nK : n ≥ K
xz : 0 < x - z
hK : (x - z)⁻¹ ≤ ↑n
n0 : 0 < n
⊢ z ≤ ↑(↑g n)

Next tactic:
refine' le_trans _ (hf₂ _ n0 _ xS).le