Step | Hyp | Ref
| Expression |
1 | | pntibnd.r |
. . 3
⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦
((ψ‘𝑎) −
𝑎)) |
2 | 1 | pntrsumbnd2 25056 |
. 2
⊢
∃𝑑 ∈
ℝ+ ∀𝑖 ∈ ℕ ∀𝑗 ∈ ℤ (abs‘Σ𝑛 ∈ (𝑖...𝑗)((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ≤ 𝑑 |
3 | | simpl 472 |
. . . . 5
⊢ ((𝑑 ∈ ℝ+
∧ ∀𝑖 ∈
ℕ ∀𝑗 ∈
ℤ (abs‘Σ𝑛
∈ (𝑖...𝑗)((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ≤ 𝑑) → 𝑑 ∈ ℝ+) |
4 | | 2rp 11713 |
. . . . 5
⊢ 2 ∈
ℝ+ |
5 | | rpaddcl 11730 |
. . . . 5
⊢ ((𝑑 ∈ ℝ+
∧ 2 ∈ ℝ+) → (𝑑 + 2) ∈
ℝ+) |
6 | 3, 4, 5 | sylancl 693 |
. . . 4
⊢ ((𝑑 ∈ ℝ+
∧ ∀𝑖 ∈
ℕ ∀𝑗 ∈
ℤ (abs‘Σ𝑛
∈ (𝑖...𝑗)((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ≤ 𝑑) → (𝑑 + 2) ∈
ℝ+) |
7 | | 2re 10967 |
. . . . . . . 8
⊢ 2 ∈
ℝ |
8 | | elioore 12076 |
. . . . . . . . . 10
⊢ (𝑒 ∈ (0(,)1) → 𝑒 ∈
ℝ) |
9 | 8 | adantl 481 |
. . . . . . . . 9
⊢ (((𝑑 ∈ ℝ+
∧ ∀𝑖 ∈
ℕ ∀𝑗 ∈
ℤ (abs‘Σ𝑛
∈ (𝑖...𝑗)((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ≤ 𝑑) ∧ 𝑒 ∈ (0(,)1)) → 𝑒 ∈ ℝ) |
10 | | eliooord 12104 |
. . . . . . . . . . 11
⊢ (𝑒 ∈ (0(,)1) → (0 <
𝑒 ∧ 𝑒 < 1)) |
11 | 10 | adantl 481 |
. . . . . . . . . 10
⊢ (((𝑑 ∈ ℝ+
∧ ∀𝑖 ∈
ℕ ∀𝑗 ∈
ℤ (abs‘Σ𝑛
∈ (𝑖...𝑗)((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ≤ 𝑑) ∧ 𝑒 ∈ (0(,)1)) → (0 < 𝑒 ∧ 𝑒 < 1)) |
12 | 11 | simpld 474 |
. . . . . . . . 9
⊢ (((𝑑 ∈ ℝ+
∧ ∀𝑖 ∈
ℕ ∀𝑗 ∈
ℤ (abs‘Σ𝑛
∈ (𝑖...𝑗)((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ≤ 𝑑) ∧ 𝑒 ∈ (0(,)1)) → 0 < 𝑒) |
13 | 9, 12 | elrpd 11745 |
. . . . . . . 8
⊢ (((𝑑 ∈ ℝ+
∧ ∀𝑖 ∈
ℕ ∀𝑗 ∈
ℤ (abs‘Σ𝑛
∈ (𝑖...𝑗)((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ≤ 𝑑) ∧ 𝑒 ∈ (0(,)1)) → 𝑒 ∈ ℝ+) |
14 | | rerpdivcl 11737 |
. . . . . . . 8
⊢ ((2
∈ ℝ ∧ 𝑒
∈ ℝ+) → (2 / 𝑒) ∈ ℝ) |
15 | 7, 13, 14 | sylancr 694 |
. . . . . . 7
⊢ (((𝑑 ∈ ℝ+
∧ ∀𝑖 ∈
ℕ ∀𝑗 ∈
ℤ (abs‘Σ𝑛
∈ (𝑖...𝑗)((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ≤ 𝑑) ∧ 𝑒 ∈ (0(,)1)) → (2 / 𝑒) ∈
ℝ) |
16 | 15 | rpefcld 14674 |
. . . . . 6
⊢ (((𝑑 ∈ ℝ+
∧ ∀𝑖 ∈
ℕ ∀𝑗 ∈
ℤ (abs‘Σ𝑛
∈ (𝑖...𝑗)((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ≤ 𝑑) ∧ 𝑒 ∈ (0(,)1)) → (exp‘(2 / 𝑒)) ∈
ℝ+) |
17 | | simpllr 795 |
. . . . . . . . 9
⊢
(((((𝑑 ∈
ℝ+ ∧ ∀𝑖 ∈ ℕ ∀𝑗 ∈ ℤ (abs‘Σ𝑛 ∈ (𝑖...𝑗)((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ≤ 𝑑) ∧ 𝑒 ∈ (0(,)1)) ∧ (𝑘 ∈ ((exp‘((𝑑 + 2) / 𝑒))[,)+∞) ∧ 𝑦 ∈ ((exp‘(2 / 𝑒))(,)+∞))) ∧ ¬ ∃𝑛 ∈ ℕ ((𝑦 < 𝑛 ∧ 𝑛 ≤ (𝑘 · 𝑦)) ∧ (abs‘((𝑅‘𝑛) / 𝑛)) ≤ 𝑒)) → 𝑒 ∈ (0(,)1)) |
18 | | eqid 2610 |
. . . . . . . . 9
⊢
(exp‘(2 / 𝑒))
= (exp‘(2 / 𝑒)) |
19 | | simplrr 797 |
. . . . . . . . 9
⊢
(((((𝑑 ∈
ℝ+ ∧ ∀𝑖 ∈ ℕ ∀𝑗 ∈ ℤ (abs‘Σ𝑛 ∈ (𝑖...𝑗)((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ≤ 𝑑) ∧ 𝑒 ∈ (0(,)1)) ∧ (𝑘 ∈ ((exp‘((𝑑 + 2) / 𝑒))[,)+∞) ∧ 𝑦 ∈ ((exp‘(2 / 𝑒))(,)+∞))) ∧ ¬ ∃𝑛 ∈ ℕ ((𝑦 < 𝑛 ∧ 𝑛 ≤ (𝑘 · 𝑦)) ∧ (abs‘((𝑅‘𝑛) / 𝑛)) ≤ 𝑒)) → 𝑦 ∈ ((exp‘(2 / 𝑒))(,)+∞)) |
20 | | simp-4l 802 |
. . . . . . . . 9
⊢
(((((𝑑 ∈
ℝ+ ∧ ∀𝑖 ∈ ℕ ∀𝑗 ∈ ℤ (abs‘Σ𝑛 ∈ (𝑖...𝑗)((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ≤ 𝑑) ∧ 𝑒 ∈ (0(,)1)) ∧ (𝑘 ∈ ((exp‘((𝑑 + 2) / 𝑒))[,)+∞) ∧ 𝑦 ∈ ((exp‘(2 / 𝑒))(,)+∞))) ∧ ¬ ∃𝑛 ∈ ℕ ((𝑦 < 𝑛 ∧ 𝑛 ≤ (𝑘 · 𝑦)) ∧ (abs‘((𝑅‘𝑛) / 𝑛)) ≤ 𝑒)) → 𝑑 ∈ ℝ+) |
21 | | simp-4r 803 |
. . . . . . . . 9
⊢
(((((𝑑 ∈
ℝ+ ∧ ∀𝑖 ∈ ℕ ∀𝑗 ∈ ℤ (abs‘Σ𝑛 ∈ (𝑖...𝑗)((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ≤ 𝑑) ∧ 𝑒 ∈ (0(,)1)) ∧ (𝑘 ∈ ((exp‘((𝑑 + 2) / 𝑒))[,)+∞) ∧ 𝑦 ∈ ((exp‘(2 / 𝑒))(,)+∞))) ∧ ¬ ∃𝑛 ∈ ℕ ((𝑦 < 𝑛 ∧ 𝑛 ≤ (𝑘 · 𝑦)) ∧ (abs‘((𝑅‘𝑛) / 𝑛)) ≤ 𝑒)) → ∀𝑖 ∈ ℕ ∀𝑗 ∈ ℤ (abs‘Σ𝑛 ∈ (𝑖...𝑗)((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ≤ 𝑑) |
22 | | eqid 2610 |
. . . . . . . . 9
⊢ (𝑑 + 2) = (𝑑 + 2) |
23 | | simplrl 796 |
. . . . . . . . 9
⊢
(((((𝑑 ∈
ℝ+ ∧ ∀𝑖 ∈ ℕ ∀𝑗 ∈ ℤ (abs‘Σ𝑛 ∈ (𝑖...𝑗)((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ≤ 𝑑) ∧ 𝑒 ∈ (0(,)1)) ∧ (𝑘 ∈ ((exp‘((𝑑 + 2) / 𝑒))[,)+∞) ∧ 𝑦 ∈ ((exp‘(2 / 𝑒))(,)+∞))) ∧ ¬ ∃𝑛 ∈ ℕ ((𝑦 < 𝑛 ∧ 𝑛 ≤ (𝑘 · 𝑦)) ∧ (abs‘((𝑅‘𝑛) / 𝑛)) ≤ 𝑒)) → 𝑘 ∈ ((exp‘((𝑑 + 2) / 𝑒))[,)+∞)) |
24 | | simpr 476 |
. . . . . . . . 9
⊢
(((((𝑑 ∈
ℝ+ ∧ ∀𝑖 ∈ ℕ ∀𝑗 ∈ ℤ (abs‘Σ𝑛 ∈ (𝑖...𝑗)((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ≤ 𝑑) ∧ 𝑒 ∈ (0(,)1)) ∧ (𝑘 ∈ ((exp‘((𝑑 + 2) / 𝑒))[,)+∞) ∧ 𝑦 ∈ ((exp‘(2 / 𝑒))(,)+∞))) ∧ ¬ ∃𝑛 ∈ ℕ ((𝑦 < 𝑛 ∧ 𝑛 ≤ (𝑘 · 𝑦)) ∧ (abs‘((𝑅‘𝑛) / 𝑛)) ≤ 𝑒)) → ¬ ∃𝑛 ∈ ℕ ((𝑦 < 𝑛 ∧ 𝑛 ≤ (𝑘 · 𝑦)) ∧ (abs‘((𝑅‘𝑛) / 𝑛)) ≤ 𝑒)) |
25 | 1, 17, 18, 19, 20, 21, 22, 23, 24 | pntpbnd2 25076 |
. . . . . . . 8
⊢ ¬
((((𝑑 ∈
ℝ+ ∧ ∀𝑖 ∈ ℕ ∀𝑗 ∈ ℤ (abs‘Σ𝑛 ∈ (𝑖...𝑗)((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ≤ 𝑑) ∧ 𝑒 ∈ (0(,)1)) ∧ (𝑘 ∈ ((exp‘((𝑑 + 2) / 𝑒))[,)+∞) ∧ 𝑦 ∈ ((exp‘(2 / 𝑒))(,)+∞))) ∧ ¬ ∃𝑛 ∈ ℕ ((𝑦 < 𝑛 ∧ 𝑛 ≤ (𝑘 · 𝑦)) ∧ (abs‘((𝑅‘𝑛) / 𝑛)) ≤ 𝑒)) |
26 | | iman 439 |
. . . . . . . 8
⊢
(((((𝑑 ∈
ℝ+ ∧ ∀𝑖 ∈ ℕ ∀𝑗 ∈ ℤ (abs‘Σ𝑛 ∈ (𝑖...𝑗)((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ≤ 𝑑) ∧ 𝑒 ∈ (0(,)1)) ∧ (𝑘 ∈ ((exp‘((𝑑 + 2) / 𝑒))[,)+∞) ∧ 𝑦 ∈ ((exp‘(2 / 𝑒))(,)+∞))) → ∃𝑛 ∈ ℕ ((𝑦 < 𝑛 ∧ 𝑛 ≤ (𝑘 · 𝑦)) ∧ (abs‘((𝑅‘𝑛) / 𝑛)) ≤ 𝑒)) ↔ ¬ ((((𝑑 ∈ ℝ+ ∧
∀𝑖 ∈ ℕ
∀𝑗 ∈ ℤ
(abs‘Σ𝑛 ∈
(𝑖...𝑗)((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ≤ 𝑑) ∧ 𝑒 ∈ (0(,)1)) ∧ (𝑘 ∈ ((exp‘((𝑑 + 2) / 𝑒))[,)+∞) ∧ 𝑦 ∈ ((exp‘(2 / 𝑒))(,)+∞))) ∧ ¬ ∃𝑛 ∈ ℕ ((𝑦 < 𝑛 ∧ 𝑛 ≤ (𝑘 · 𝑦)) ∧ (abs‘((𝑅‘𝑛) / 𝑛)) ≤ 𝑒))) |
27 | 25, 26 | mpbir 220 |
. . . . . . 7
⊢ ((((𝑑 ∈ ℝ+
∧ ∀𝑖 ∈
ℕ ∀𝑗 ∈
ℤ (abs‘Σ𝑛
∈ (𝑖...𝑗)((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ≤ 𝑑) ∧ 𝑒 ∈ (0(,)1)) ∧ (𝑘 ∈ ((exp‘((𝑑 + 2) / 𝑒))[,)+∞) ∧ 𝑦 ∈ ((exp‘(2 / 𝑒))(,)+∞))) → ∃𝑛 ∈ ℕ ((𝑦 < 𝑛 ∧ 𝑛 ≤ (𝑘 · 𝑦)) ∧ (abs‘((𝑅‘𝑛) / 𝑛)) ≤ 𝑒)) |
28 | 27 | ralrimivva 2954 |
. . . . . 6
⊢ (((𝑑 ∈ ℝ+
∧ ∀𝑖 ∈
ℕ ∀𝑗 ∈
ℤ (abs‘Σ𝑛
∈ (𝑖...𝑗)((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ≤ 𝑑) ∧ 𝑒 ∈ (0(,)1)) → ∀𝑘 ∈ ((exp‘((𝑑 + 2) / 𝑒))[,)+∞)∀𝑦 ∈ ((exp‘(2 / 𝑒))(,)+∞)∃𝑛 ∈ ℕ ((𝑦 < 𝑛 ∧ 𝑛 ≤ (𝑘 · 𝑦)) ∧ (abs‘((𝑅‘𝑛) / 𝑛)) ≤ 𝑒)) |
29 | | oveq1 6556 |
. . . . . . . . 9
⊢ (𝑥 = (exp‘(2 / 𝑒)) → (𝑥(,)+∞) = ((exp‘(2 / 𝑒))(,)+∞)) |
30 | 29 | raleqdv 3121 |
. . . . . . . 8
⊢ (𝑥 = (exp‘(2 / 𝑒)) → (∀𝑦 ∈ (𝑥(,)+∞)∃𝑛 ∈ ℕ ((𝑦 < 𝑛 ∧ 𝑛 ≤ (𝑘 · 𝑦)) ∧ (abs‘((𝑅‘𝑛) / 𝑛)) ≤ 𝑒) ↔ ∀𝑦 ∈ ((exp‘(2 / 𝑒))(,)+∞)∃𝑛 ∈ ℕ ((𝑦 < 𝑛 ∧ 𝑛 ≤ (𝑘 · 𝑦)) ∧ (abs‘((𝑅‘𝑛) / 𝑛)) ≤ 𝑒))) |
31 | 30 | ralbidv 2969 |
. . . . . . 7
⊢ (𝑥 = (exp‘(2 / 𝑒)) → (∀𝑘 ∈ ((exp‘((𝑑 + 2) / 𝑒))[,)+∞)∀𝑦 ∈ (𝑥(,)+∞)∃𝑛 ∈ ℕ ((𝑦 < 𝑛 ∧ 𝑛 ≤ (𝑘 · 𝑦)) ∧ (abs‘((𝑅‘𝑛) / 𝑛)) ≤ 𝑒) ↔ ∀𝑘 ∈ ((exp‘((𝑑 + 2) / 𝑒))[,)+∞)∀𝑦 ∈ ((exp‘(2 / 𝑒))(,)+∞)∃𝑛 ∈ ℕ ((𝑦 < 𝑛 ∧ 𝑛 ≤ (𝑘 · 𝑦)) ∧ (abs‘((𝑅‘𝑛) / 𝑛)) ≤ 𝑒))) |
32 | 31 | rspcev 3282 |
. . . . . 6
⊢
(((exp‘(2 / 𝑒)) ∈ ℝ+ ∧
∀𝑘 ∈
((exp‘((𝑑 + 2) /
𝑒))[,)+∞)∀𝑦 ∈ ((exp‘(2 / 𝑒))(,)+∞)∃𝑛 ∈ ℕ ((𝑦 < 𝑛 ∧ 𝑛 ≤ (𝑘 · 𝑦)) ∧ (abs‘((𝑅‘𝑛) / 𝑛)) ≤ 𝑒)) → ∃𝑥 ∈ ℝ+ ∀𝑘 ∈ ((exp‘((𝑑 + 2) / 𝑒))[,)+∞)∀𝑦 ∈ (𝑥(,)+∞)∃𝑛 ∈ ℕ ((𝑦 < 𝑛 ∧ 𝑛 ≤ (𝑘 · 𝑦)) ∧ (abs‘((𝑅‘𝑛) / 𝑛)) ≤ 𝑒)) |
33 | 16, 28, 32 | syl2anc 691 |
. . . . 5
⊢ (((𝑑 ∈ ℝ+
∧ ∀𝑖 ∈
ℕ ∀𝑗 ∈
ℤ (abs‘Σ𝑛
∈ (𝑖...𝑗)((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ≤ 𝑑) ∧ 𝑒 ∈ (0(,)1)) → ∃𝑥 ∈ ℝ+
∀𝑘 ∈
((exp‘((𝑑 + 2) /
𝑒))[,)+∞)∀𝑦 ∈ (𝑥(,)+∞)∃𝑛 ∈ ℕ ((𝑦 < 𝑛 ∧ 𝑛 ≤ (𝑘 · 𝑦)) ∧ (abs‘((𝑅‘𝑛) / 𝑛)) ≤ 𝑒)) |
34 | 33 | ralrimiva 2949 |
. . . 4
⊢ ((𝑑 ∈ ℝ+
∧ ∀𝑖 ∈
ℕ ∀𝑗 ∈
ℤ (abs‘Σ𝑛
∈ (𝑖...𝑗)((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ≤ 𝑑) → ∀𝑒 ∈ (0(,)1)∃𝑥 ∈ ℝ+ ∀𝑘 ∈ ((exp‘((𝑑 + 2) / 𝑒))[,)+∞)∀𝑦 ∈ (𝑥(,)+∞)∃𝑛 ∈ ℕ ((𝑦 < 𝑛 ∧ 𝑛 ≤ (𝑘 · 𝑦)) ∧ (abs‘((𝑅‘𝑛) / 𝑛)) ≤ 𝑒)) |
35 | | oveq1 6556 |
. . . . . . . . . 10
⊢ (𝑐 = (𝑑 + 2) → (𝑐 / 𝑒) = ((𝑑 + 2) / 𝑒)) |
36 | 35 | fveq2d 6107 |
. . . . . . . . 9
⊢ (𝑐 = (𝑑 + 2) → (exp‘(𝑐 / 𝑒)) = (exp‘((𝑑 + 2) / 𝑒))) |
37 | 36 | oveq1d 6564 |
. . . . . . . 8
⊢ (𝑐 = (𝑑 + 2) → ((exp‘(𝑐 / 𝑒))[,)+∞) = ((exp‘((𝑑 + 2) / 𝑒))[,)+∞)) |
38 | 37 | raleqdv 3121 |
. . . . . . 7
⊢ (𝑐 = (𝑑 + 2) → (∀𝑘 ∈ ((exp‘(𝑐 / 𝑒))[,)+∞)∀𝑦 ∈ (𝑥(,)+∞)∃𝑛 ∈ ℕ ((𝑦 < 𝑛 ∧ 𝑛 ≤ (𝑘 · 𝑦)) ∧ (abs‘((𝑅‘𝑛) / 𝑛)) ≤ 𝑒) ↔ ∀𝑘 ∈ ((exp‘((𝑑 + 2) / 𝑒))[,)+∞)∀𝑦 ∈ (𝑥(,)+∞)∃𝑛 ∈ ℕ ((𝑦 < 𝑛 ∧ 𝑛 ≤ (𝑘 · 𝑦)) ∧ (abs‘((𝑅‘𝑛) / 𝑛)) ≤ 𝑒))) |
39 | 38 | rexbidv 3034 |
. . . . . 6
⊢ (𝑐 = (𝑑 + 2) → (∃𝑥 ∈ ℝ+ ∀𝑘 ∈ ((exp‘(𝑐 / 𝑒))[,)+∞)∀𝑦 ∈ (𝑥(,)+∞)∃𝑛 ∈ ℕ ((𝑦 < 𝑛 ∧ 𝑛 ≤ (𝑘 · 𝑦)) ∧ (abs‘((𝑅‘𝑛) / 𝑛)) ≤ 𝑒) ↔ ∃𝑥 ∈ ℝ+ ∀𝑘 ∈ ((exp‘((𝑑 + 2) / 𝑒))[,)+∞)∀𝑦 ∈ (𝑥(,)+∞)∃𝑛 ∈ ℕ ((𝑦 < 𝑛 ∧ 𝑛 ≤ (𝑘 · 𝑦)) ∧ (abs‘((𝑅‘𝑛) / 𝑛)) ≤ 𝑒))) |
40 | 39 | ralbidv 2969 |
. . . . 5
⊢ (𝑐 = (𝑑 + 2) → (∀𝑒 ∈ (0(,)1)∃𝑥 ∈ ℝ+ ∀𝑘 ∈ ((exp‘(𝑐 / 𝑒))[,)+∞)∀𝑦 ∈ (𝑥(,)+∞)∃𝑛 ∈ ℕ ((𝑦 < 𝑛 ∧ 𝑛 ≤ (𝑘 · 𝑦)) ∧ (abs‘((𝑅‘𝑛) / 𝑛)) ≤ 𝑒) ↔ ∀𝑒 ∈ (0(,)1)∃𝑥 ∈ ℝ+ ∀𝑘 ∈ ((exp‘((𝑑 + 2) / 𝑒))[,)+∞)∀𝑦 ∈ (𝑥(,)+∞)∃𝑛 ∈ ℕ ((𝑦 < 𝑛 ∧ 𝑛 ≤ (𝑘 · 𝑦)) ∧ (abs‘((𝑅‘𝑛) / 𝑛)) ≤ 𝑒))) |
41 | 40 | rspcev 3282 |
. . . 4
⊢ (((𝑑 + 2) ∈ ℝ+
∧ ∀𝑒 ∈
(0(,)1)∃𝑥 ∈
ℝ+ ∀𝑘 ∈ ((exp‘((𝑑 + 2) / 𝑒))[,)+∞)∀𝑦 ∈ (𝑥(,)+∞)∃𝑛 ∈ ℕ ((𝑦 < 𝑛 ∧ 𝑛 ≤ (𝑘 · 𝑦)) ∧ (abs‘((𝑅‘𝑛) / 𝑛)) ≤ 𝑒)) → ∃𝑐 ∈ ℝ+ ∀𝑒 ∈ (0(,)1)∃𝑥 ∈ ℝ+
∀𝑘 ∈
((exp‘(𝑐 / 𝑒))[,)+∞)∀𝑦 ∈ (𝑥(,)+∞)∃𝑛 ∈ ℕ ((𝑦 < 𝑛 ∧ 𝑛 ≤ (𝑘 · 𝑦)) ∧ (abs‘((𝑅‘𝑛) / 𝑛)) ≤ 𝑒)) |
42 | 6, 34, 41 | syl2anc 691 |
. . 3
⊢ ((𝑑 ∈ ℝ+
∧ ∀𝑖 ∈
ℕ ∀𝑗 ∈
ℤ (abs‘Σ𝑛
∈ (𝑖...𝑗)((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ≤ 𝑑) → ∃𝑐 ∈ ℝ+ ∀𝑒 ∈ (0(,)1)∃𝑥 ∈ ℝ+
∀𝑘 ∈
((exp‘(𝑐 / 𝑒))[,)+∞)∀𝑦 ∈ (𝑥(,)+∞)∃𝑛 ∈ ℕ ((𝑦 < 𝑛 ∧ 𝑛 ≤ (𝑘 · 𝑦)) ∧ (abs‘((𝑅‘𝑛) / 𝑛)) ≤ 𝑒)) |
43 | 42 | rexlimiva 3010 |
. 2
⊢
(∃𝑑 ∈
ℝ+ ∀𝑖 ∈ ℕ ∀𝑗 ∈ ℤ (abs‘Σ𝑛 ∈ (𝑖...𝑗)((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ≤ 𝑑 → ∃𝑐 ∈ ℝ+ ∀𝑒 ∈ (0(,)1)∃𝑥 ∈ ℝ+
∀𝑘 ∈
((exp‘(𝑐 / 𝑒))[,)+∞)∀𝑦 ∈ (𝑥(,)+∞)∃𝑛 ∈ ℕ ((𝑦 < 𝑛 ∧ 𝑛 ≤ (𝑘 · 𝑦)) ∧ (abs‘((𝑅‘𝑛) / 𝑛)) ≤ 𝑒)) |
44 | 2, 43 | ax-mp 5 |
1
⊢
∃𝑐 ∈
ℝ+ ∀𝑒 ∈ (0(,)1)∃𝑥 ∈ ℝ+ ∀𝑘 ∈ ((exp‘(𝑐 / 𝑒))[,)+∞)∀𝑦 ∈ (𝑥(,)+∞)∃𝑛 ∈ ℕ ((𝑦 < 𝑛 ∧ 𝑛 ≤ (𝑘 · 𝑦)) ∧ (abs‘((𝑅‘𝑛) / 𝑛)) ≤ 𝑒) |