statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
geom_le {u : ℕ → ℝ} {c : ℝ} (hc : 0 ≤ c) (n : ℕ) (h : ∀ k < n, c * u k ≤ u (k + 1)) :
c ^ n * u 0 ≤ u n | by refine (monotone_mul_left_of_nonneg hc).seq_le_seq n _ _ h; simp [pow_succ, mul_assoc, le_refl] | lemma | geom_le | analysis.specific_limits | src/analysis/specific_limits/basic.lean | [
"algebra.geom_sum",
"order.filter.archimedean",
"order.iterate",
"topology.instances.ennreal",
"topology.algebra.algebra"
] | [
"monotone_mul_left_of_nonneg",
"mul_assoc",
"pow_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_geom {u : ℕ → ℝ} {c : ℝ} (hc : 0 ≤ c) {n : ℕ} (hn : 0 < n)
(h : ∀ k < n, u (k + 1) < c * u k) :
u n < c ^ n * u 0 | begin
refine (monotone_mul_left_of_nonneg hc).seq_pos_lt_seq_of_lt_of_le hn _ h _,
{ simp },
{ simp [pow_succ, mul_assoc, le_refl] }
end | lemma | lt_geom | analysis.specific_limits | src/analysis/specific_limits/basic.lean | [
"algebra.geom_sum",
"order.filter.archimedean",
"order.iterate",
"topology.instances.ennreal",
"topology.algebra.algebra"
] | [
"monotone_mul_left_of_nonneg",
"mul_assoc",
"pow_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_geom {u : ℕ → ℝ} {c : ℝ} (hc : 0 ≤ c) (n : ℕ) (h : ∀ k < n, u (k + 1) ≤ c * u k) :
u n ≤ (c ^ n) * u 0 | by refine (monotone_mul_left_of_nonneg hc).seq_le_seq n _ h _; simp [pow_succ, mul_assoc, le_refl] | lemma | le_geom | analysis.specific_limits | src/analysis/specific_limits/basic.lean | [
"algebra.geom_sum",
"order.filter.archimedean",
"order.iterate",
"topology.instances.ennreal",
"topology.algebra.algebra"
] | [
"monotone_mul_left_of_nonneg",
"mul_assoc",
"pow_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_at_top_of_geom_le {v : ℕ → ℝ} {c : ℝ} (h₀ : 0 < v 0) (hc : 1 < c)
(hu : ∀ n, c * v n ≤ v (n + 1)) : tendsto v at_top at_top | tendsto_at_top_mono (λ n, geom_le (zero_le_one.trans hc.le) n (λ k hk, hu k)) $
(tendsto_pow_at_top_at_top_of_one_lt hc).at_top_mul_const h₀ | lemma | tendsto_at_top_of_geom_le | analysis.specific_limits | src/analysis/specific_limits/basic.lean | [
"algebra.geom_sum",
"order.filter.archimedean",
"order.iterate",
"topology.instances.ennreal",
"topology.algebra.algebra"
] | [
"geom_le",
"tendsto_pow_at_top_at_top_of_one_lt"
] | If a sequence `v` of real numbers satisfies `k * v n ≤ v (n+1)` with `1 < k`,
then it goes to +∞. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nnreal.tendsto_pow_at_top_nhds_0_of_lt_1 {r : ℝ≥0} (hr : r < 1) :
tendsto (λ n:ℕ, r^n) at_top (𝓝 0) | nnreal.tendsto_coe.1 $ by simp only [nnreal.coe_pow, nnreal.coe_zero,
tendsto_pow_at_top_nhds_0_of_lt_1 r.coe_nonneg hr] | lemma | nnreal.tendsto_pow_at_top_nhds_0_of_lt_1 | analysis.specific_limits | src/analysis/specific_limits/basic.lean | [
"algebra.geom_sum",
"order.filter.archimedean",
"order.iterate",
"topology.instances.ennreal",
"topology.algebra.algebra"
] | [
"nnreal.coe_pow",
"nnreal.coe_zero",
"tendsto_pow_at_top_nhds_0_of_lt_1"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ennreal.tendsto_pow_at_top_nhds_0_of_lt_1 {r : ℝ≥0∞} (hr : r < 1) :
tendsto (λ n:ℕ, r^n) at_top (𝓝 0) | begin
rcases ennreal.lt_iff_exists_coe.1 hr with ⟨r, rfl, hr'⟩,
rw [← ennreal.coe_zero],
norm_cast at *,
apply nnreal.tendsto_pow_at_top_nhds_0_of_lt_1 hr
end | lemma | ennreal.tendsto_pow_at_top_nhds_0_of_lt_1 | analysis.specific_limits | src/analysis/specific_limits/basic.lean | [
"algebra.geom_sum",
"order.filter.archimedean",
"order.iterate",
"topology.instances.ennreal",
"topology.algebra.algebra"
] | [
"ennreal.coe_zero",
"nnreal.tendsto_pow_at_top_nhds_0_of_lt_1"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_geometric_of_lt_1 {r : ℝ} (h₁ : 0 ≤ r) (h₂ : r < 1) :
has_sum (λn:ℕ, r ^ n) (1 - r)⁻¹ | have r ≠ 1, from ne_of_lt h₂,
have tendsto (λn, (r ^ n - 1) * (r - 1)⁻¹) at_top (𝓝 ((0 - 1) * (r - 1)⁻¹)),
from ((tendsto_pow_at_top_nhds_0_of_lt_1 h₁ h₂).sub tendsto_const_nhds).mul tendsto_const_nhds,
(has_sum_iff_tendsto_nat_of_nonneg (pow_nonneg h₁) _).mpr $
by simp [neg_inv, geom_sum_eq, div_eq_mul_inv, *] at... | lemma | has_sum_geometric_of_lt_1 | analysis.specific_limits | src/analysis/specific_limits/basic.lean | [
"algebra.geom_sum",
"order.filter.archimedean",
"order.iterate",
"topology.instances.ennreal",
"topology.algebra.algebra"
] | [
"div_eq_mul_inv",
"geom_sum_eq",
"has_sum",
"has_sum_iff_tendsto_nat_of_nonneg",
"neg_inv",
"pow_nonneg",
"tendsto_const_nhds",
"tendsto_pow_at_top_nhds_0_of_lt_1"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_geometric_of_lt_1 {r : ℝ} (h₁ : 0 ≤ r) (h₂ : r < 1) : summable (λn:ℕ, r ^ n) | ⟨_, has_sum_geometric_of_lt_1 h₁ h₂⟩ | lemma | summable_geometric_of_lt_1 | analysis.specific_limits | src/analysis/specific_limits/basic.lean | [
"algebra.geom_sum",
"order.filter.archimedean",
"order.iterate",
"topology.instances.ennreal",
"topology.algebra.algebra"
] | [
"has_sum_geometric_of_lt_1",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_geometric_of_lt_1 {r : ℝ} (h₁ : 0 ≤ r) (h₂ : r < 1) : ∑'n:ℕ, r ^ n = (1 - r)⁻¹ | (has_sum_geometric_of_lt_1 h₁ h₂).tsum_eq | lemma | tsum_geometric_of_lt_1 | analysis.specific_limits | src/analysis/specific_limits/basic.lean | [
"algebra.geom_sum",
"order.filter.archimedean",
"order.iterate",
"topology.instances.ennreal",
"topology.algebra.algebra"
] | [
"has_sum_geometric_of_lt_1"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_geometric_two : has_sum (λn:ℕ, ((1:ℝ)/2) ^ n) 2 | by convert has_sum_geometric_of_lt_1 _ _; norm_num | lemma | has_sum_geometric_two | analysis.specific_limits | src/analysis/specific_limits/basic.lean | [
"algebra.geom_sum",
"order.filter.archimedean",
"order.iterate",
"topology.instances.ennreal",
"topology.algebra.algebra"
] | [
"has_sum",
"has_sum_geometric_of_lt_1"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_geometric_two : summable (λn:ℕ, ((1:ℝ)/2) ^ n) | ⟨_, has_sum_geometric_two⟩ | lemma | summable_geometric_two | analysis.specific_limits | src/analysis/specific_limits/basic.lean | [
"algebra.geom_sum",
"order.filter.archimedean",
"order.iterate",
"topology.instances.ennreal",
"topology.algebra.algebra"
] | [
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_geometric_two_encode {ι : Type*} [encodable ι] :
summable (λ (i : ι), (1/2 : ℝ)^(encodable.encode i)) | summable_geometric_two.comp_injective encodable.encode_injective | lemma | summable_geometric_two_encode | analysis.specific_limits | src/analysis/specific_limits/basic.lean | [
"algebra.geom_sum",
"order.filter.archimedean",
"order.iterate",
"topology.instances.ennreal",
"topology.algebra.algebra"
] | [
"encodable",
"encodable.encode_injective",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_geometric_two : ∑'n:ℕ, ((1:ℝ)/2) ^ n = 2 | has_sum_geometric_two.tsum_eq | lemma | tsum_geometric_two | analysis.specific_limits | src/analysis/specific_limits/basic.lean | [
"algebra.geom_sum",
"order.filter.archimedean",
"order.iterate",
"topology.instances.ennreal",
"topology.algebra.algebra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_geometric_two_le (n : ℕ) : ∑ (i : ℕ) in range n, (1 / (2 : ℝ)) ^ i ≤ 2 | begin
have : ∀ i, 0 ≤ (1 / (2 : ℝ)) ^ i,
{ intro i, apply pow_nonneg, norm_num },
convert sum_le_tsum (range n) (λ i _, this i) summable_geometric_two,
exact tsum_geometric_two.symm
end | lemma | sum_geometric_two_le | analysis.specific_limits | src/analysis/specific_limits/basic.lean | [
"algebra.geom_sum",
"order.filter.archimedean",
"order.iterate",
"topology.instances.ennreal",
"topology.algebra.algebra"
] | [
"pow_nonneg",
"sum_le_tsum",
"summable_geometric_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_geometric_inv_two : ∑' n : ℕ, (2 : ℝ)⁻¹ ^ n = 2 | (inv_eq_one_div (2 : ℝ)).symm ▸ tsum_geometric_two | lemma | tsum_geometric_inv_two | analysis.specific_limits | src/analysis/specific_limits/basic.lean | [
"algebra.geom_sum",
"order.filter.archimedean",
"order.iterate",
"topology.instances.ennreal",
"topology.algebra.algebra"
] | [
"inv_eq_one_div",
"tsum_geometric_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_geometric_inv_two_ge (n : ℕ) :
∑' i, ite (n ≤ i) ((2 : ℝ)⁻¹ ^ i) 0 = 2 * 2⁻¹ ^ n | begin
have A : summable (λ (i : ℕ), ite (n ≤ i) ((2⁻¹ : ℝ) ^ i) 0),
{ apply summable_of_nonneg_of_le _ _ summable_geometric_two;
{ intro i, by_cases hi : n ≤ i; simp [hi] } },
have B : (finset.range n).sum (λ (i : ℕ), ite (n ≤ i) ((2⁻¹ : ℝ)^i) 0) = 0 :=
finset.sum_eq_zero (λ i hi, ite_eq_right_iff.2 $ λ h... | lemma | tsum_geometric_inv_two_ge | analysis.specific_limits | src/analysis/specific_limits/basic.lean | [
"algebra.geom_sum",
"order.filter.archimedean",
"order.iterate",
"topology.instances.ennreal",
"topology.algebra.algebra"
] | [
"finset.range",
"pow_add",
"sum_add_tsum_nat_add",
"summable",
"summable_geometric_two",
"summable_of_nonneg_of_le",
"tsum_geometric_inv_two",
"tsum_mul_right",
"zero_le'"
] | The sum of `2⁻¹ ^ i` for `n ≤ i` equals `2 * 2⁻¹ ^ n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_sum_geometric_two' (a : ℝ) : has_sum (λn:ℕ, (a / 2) / 2 ^ n) a | begin
convert has_sum.mul_left (a / 2) (has_sum_geometric_of_lt_1
(le_of_lt one_half_pos) one_half_lt_one),
{ funext n, simp, refl, },
{ norm_num }
end | lemma | has_sum_geometric_two' | analysis.specific_limits | src/analysis/specific_limits/basic.lean | [
"algebra.geom_sum",
"order.filter.archimedean",
"order.iterate",
"topology.instances.ennreal",
"topology.algebra.algebra"
] | [
"has_sum",
"has_sum.mul_left",
"has_sum_geometric_of_lt_1",
"one_half_lt_one",
"one_half_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_geometric_two' (a : ℝ) : summable (λ n:ℕ, (a / 2) / 2 ^ n) | ⟨a, has_sum_geometric_two' a⟩ | lemma | summable_geometric_two' | analysis.specific_limits | src/analysis/specific_limits/basic.lean | [
"algebra.geom_sum",
"order.filter.archimedean",
"order.iterate",
"topology.instances.ennreal",
"topology.algebra.algebra"
] | [
"has_sum_geometric_two'",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_geometric_two' (a : ℝ) : ∑' n:ℕ, (a / 2) / 2^n = a | (has_sum_geometric_two' a).tsum_eq | lemma | tsum_geometric_two' | analysis.specific_limits | src/analysis/specific_limits/basic.lean | [
"algebra.geom_sum",
"order.filter.archimedean",
"order.iterate",
"topology.instances.ennreal",
"topology.algebra.algebra"
] | [
"has_sum_geometric_two'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnreal.has_sum_geometric {r : ℝ≥0} (hr : r < 1) :
has_sum (λ n : ℕ, r ^ n) (1 - r)⁻¹ | begin
apply nnreal.has_sum_coe.1,
push_cast,
rw [nnreal.coe_sub (le_of_lt hr)],
exact has_sum_geometric_of_lt_1 r.coe_nonneg hr
end | lemma | nnreal.has_sum_geometric | analysis.specific_limits | src/analysis/specific_limits/basic.lean | [
"algebra.geom_sum",
"order.filter.archimedean",
"order.iterate",
"topology.instances.ennreal",
"topology.algebra.algebra"
] | [
"has_sum",
"has_sum_geometric_of_lt_1",
"nnreal.coe_sub"
] | **Sum of a Geometric Series** | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nnreal.summable_geometric {r : ℝ≥0} (hr : r < 1) : summable (λn:ℕ, r ^ n) | ⟨_, nnreal.has_sum_geometric hr⟩ | lemma | nnreal.summable_geometric | analysis.specific_limits | src/analysis/specific_limits/basic.lean | [
"algebra.geom_sum",
"order.filter.archimedean",
"order.iterate",
"topology.instances.ennreal",
"topology.algebra.algebra"
] | [
"nnreal.has_sum_geometric",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_geometric_nnreal {r : ℝ≥0} (hr : r < 1) : ∑'n:ℕ, r ^ n = (1 - r)⁻¹ | (nnreal.has_sum_geometric hr).tsum_eq | lemma | tsum_geometric_nnreal | analysis.specific_limits | src/analysis/specific_limits/basic.lean | [
"algebra.geom_sum",
"order.filter.archimedean",
"order.iterate",
"topology.instances.ennreal",
"topology.algebra.algebra"
] | [
"nnreal.has_sum_geometric"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ennreal.tsum_geometric (r : ℝ≥0∞) : ∑'n:ℕ, r ^ n = (1 - r)⁻¹ | begin
cases lt_or_le r 1 with hr hr,
{ rcases ennreal.lt_iff_exists_coe.1 hr with ⟨r, rfl, hr'⟩,
norm_cast at *,
convert ennreal.tsum_coe_eq (nnreal.has_sum_geometric hr),
rw [ennreal.coe_inv $ ne_of_gt $ tsub_pos_iff_lt.2 hr] },
{ rw [tsub_eq_zero_iff_le.mpr hr, ennreal.inv_zero, ennreal.tsum_eq_supr... | lemma | ennreal.tsum_geometric | analysis.specific_limits | src/analysis/specific_limits/basic.lean | [
"algebra.geom_sum",
"order.filter.archimedean",
"order.iterate",
"topology.instances.ennreal",
"topology.algebra.algebra"
] | [
"ennreal.coe_inv",
"ennreal.exists_nat_gt",
"ennreal.inv_zero",
"ennreal.tsum_coe_eq",
"ennreal.tsum_eq_supr_nat",
"nnreal.has_sum_geometric",
"nsmul_one",
"one_le_pow_of_one_le'",
"supr_eq_top"
] | The series `pow r` converges to `(1-r)⁻¹`. For `r < 1` the RHS is a finite number,
and for `1 ≤ r` the RHS equals `∞`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cauchy_seq_of_edist_le_geometric : cauchy_seq f | begin
refine cauchy_seq_of_edist_le_of_tsum_ne_top _ hu _,
rw [ennreal.tsum_mul_left, ennreal.tsum_geometric],
refine ennreal.mul_ne_top hC (ennreal.inv_ne_top.2 _),
exact (tsub_pos_iff_lt.2 hr).ne'
end | lemma | cauchy_seq_of_edist_le_geometric | analysis.specific_limits | src/analysis/specific_limits/basic.lean | [
"algebra.geom_sum",
"order.filter.archimedean",
"order.iterate",
"topology.instances.ennreal",
"topology.algebra.algebra"
] | [
"cauchy_seq",
"cauchy_seq_of_edist_le_of_tsum_ne_top",
"ennreal.mul_ne_top",
"ennreal.tsum_geometric",
"ennreal.tsum_mul_left"
] | If `edist (f n) (f (n+1))` is bounded by `C * r^n`, `C ≠ ∞`, `r < 1`,
then `f` is a Cauchy sequence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
edist_le_of_edist_le_geometric_of_tendsto {a : α} (ha : tendsto f at_top (𝓝 a)) (n : ℕ) :
edist (f n) a ≤ (C * r^n) / (1 - r) | begin
convert edist_le_tsum_of_edist_le_of_tendsto _ hu ha _,
simp only [pow_add, ennreal.tsum_mul_left, ennreal.tsum_geometric, div_eq_mul_inv, mul_assoc]
end | lemma | edist_le_of_edist_le_geometric_of_tendsto | analysis.specific_limits | src/analysis/specific_limits/basic.lean | [
"algebra.geom_sum",
"order.filter.archimedean",
"order.iterate",
"topology.instances.ennreal",
"topology.algebra.algebra"
] | [
"div_eq_mul_inv",
"edist_le_tsum_of_edist_le_of_tendsto",
"ennreal.tsum_geometric",
"ennreal.tsum_mul_left",
"mul_assoc",
"pow_add"
] | If `edist (f n) (f (n+1))` is bounded by `C * r^n`, then the distance from
`f n` to the limit of `f` is bounded above by `C * r^n / (1 - r)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
edist_le_of_edist_le_geometric_of_tendsto₀ {a : α} (ha : tendsto f at_top (𝓝 a)) :
edist (f 0) a ≤ C / (1 - r) | by simpa only [pow_zero, mul_one] using edist_le_of_edist_le_geometric_of_tendsto r C hu ha 0 | lemma | edist_le_of_edist_le_geometric_of_tendsto₀ | analysis.specific_limits | src/analysis/specific_limits/basic.lean | [
"algebra.geom_sum",
"order.filter.archimedean",
"order.iterate",
"topology.instances.ennreal",
"topology.algebra.algebra"
] | [
"edist_le_of_edist_le_geometric_of_tendsto",
"mul_one",
"pow_zero"
] | If `edist (f n) (f (n+1))` is bounded by `C * r^n`, then the distance from
`f 0` to the limit of `f` is bounded above by `C / (1 - r)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cauchy_seq_of_edist_le_geometric_two : cauchy_seq f | begin
simp only [div_eq_mul_inv, ennreal.inv_pow] at hu,
refine cauchy_seq_of_edist_le_geometric 2⁻¹ C _ hC hu,
simp [ennreal.one_lt_two]
end | lemma | cauchy_seq_of_edist_le_geometric_two | analysis.specific_limits | src/analysis/specific_limits/basic.lean | [
"algebra.geom_sum",
"order.filter.archimedean",
"order.iterate",
"topology.instances.ennreal",
"topology.algebra.algebra"
] | [
"cauchy_seq",
"cauchy_seq_of_edist_le_geometric",
"div_eq_mul_inv",
"ennreal.inv_pow",
"ennreal.one_lt_two"
] | If `edist (f n) (f (n+1))` is bounded by `C * 2^-n`, then `f` is a Cauchy sequence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
edist_le_of_edist_le_geometric_two_of_tendsto (n : ℕ) :
edist (f n) a ≤ 2 * C / 2^n | begin
simp only [div_eq_mul_inv, ennreal.inv_pow] at *,
rw [mul_assoc, mul_comm],
convert edist_le_of_edist_le_geometric_of_tendsto 2⁻¹ C hu ha n,
rw [ennreal.one_sub_inv_two, inv_inv]
end | lemma | edist_le_of_edist_le_geometric_two_of_tendsto | analysis.specific_limits | src/analysis/specific_limits/basic.lean | [
"algebra.geom_sum",
"order.filter.archimedean",
"order.iterate",
"topology.instances.ennreal",
"topology.algebra.algebra"
] | [
"div_eq_mul_inv",
"edist_le_of_edist_le_geometric_of_tendsto",
"ennreal.inv_pow",
"ennreal.one_sub_inv_two",
"inv_inv",
"mul_assoc",
"mul_comm"
] | If `edist (f n) (f (n+1))` is bounded by `C * 2^-n`, then the distance from
`f n` to the limit of `f` is bounded above by `2 * C * 2^-n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
edist_le_of_edist_le_geometric_two_of_tendsto₀: edist (f 0) a ≤ 2 * C | by simpa only [pow_zero, div_eq_mul_inv, inv_one, mul_one]
using edist_le_of_edist_le_geometric_two_of_tendsto C hu ha 0 | lemma | edist_le_of_edist_le_geometric_two_of_tendsto₀ | analysis.specific_limits | src/analysis/specific_limits/basic.lean | [
"algebra.geom_sum",
"order.filter.archimedean",
"order.iterate",
"topology.instances.ennreal",
"topology.algebra.algebra"
] | [
"div_eq_mul_inv",
"edist_le_of_edist_le_geometric_two_of_tendsto",
"inv_one",
"mul_one",
"pow_zero"
] | If `edist (f n) (f (n+1))` is bounded by `C * 2^-n`, then the distance from
`f 0` to the limit of `f` is bounded above by `2 * C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
aux_has_sum_of_le_geometric : has_sum (λ n : ℕ, C * r^n) (C / (1 - r)) | begin
rcases sign_cases_of_C_mul_pow_nonneg (λ n, dist_nonneg.trans (hu n)) with rfl | ⟨C₀, r₀⟩,
{ simp [has_sum_zero] },
{ refine has_sum.mul_left C _,
simpa using has_sum_geometric_of_lt_1 r₀ hr }
end | lemma | aux_has_sum_of_le_geometric | analysis.specific_limits | src/analysis/specific_limits/basic.lean | [
"algebra.geom_sum",
"order.filter.archimedean",
"order.iterate",
"topology.instances.ennreal",
"topology.algebra.algebra"
] | [
"has_sum",
"has_sum.mul_left",
"has_sum_geometric_of_lt_1",
"has_sum_zero",
"sign_cases_of_C_mul_pow_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cauchy_seq_of_le_geometric : cauchy_seq f | cauchy_seq_of_dist_le_of_summable _ hu ⟨_, aux_has_sum_of_le_geometric hr hu⟩ | lemma | cauchy_seq_of_le_geometric | analysis.specific_limits | src/analysis/specific_limits/basic.lean | [
"algebra.geom_sum",
"order.filter.archimedean",
"order.iterate",
"topology.instances.ennreal",
"topology.algebra.algebra"
] | [
"aux_has_sum_of_le_geometric",
"cauchy_seq",
"cauchy_seq_of_dist_le_of_summable"
] | If `dist (f n) (f (n+1))` is bounded by `C * r^n`, `r < 1`, then `f` is a Cauchy sequence.
Note that this lemma does not assume `0 ≤ C` or `0 ≤ r`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dist_le_of_le_geometric_of_tendsto₀ {a : α} (ha : tendsto f at_top (𝓝 a)) :
dist (f 0) a ≤ C / (1 - r) | (aux_has_sum_of_le_geometric hr hu).tsum_eq ▸
dist_le_tsum_of_dist_le_of_tendsto₀ _ hu ⟨_, aux_has_sum_of_le_geometric hr hu⟩ ha | lemma | dist_le_of_le_geometric_of_tendsto₀ | analysis.specific_limits | src/analysis/specific_limits/basic.lean | [
"algebra.geom_sum",
"order.filter.archimedean",
"order.iterate",
"topology.instances.ennreal",
"topology.algebra.algebra"
] | [
"aux_has_sum_of_le_geometric",
"dist_le_tsum_of_dist_le_of_tendsto₀"
] | If `dist (f n) (f (n+1))` is bounded by `C * r^n`, `r < 1`, then the distance from
`f n` to the limit of `f` is bounded above by `C * r^n / (1 - r)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dist_le_of_le_geometric_of_tendsto {a : α} (ha : tendsto f at_top (𝓝 a)) (n : ℕ) :
dist (f n) a ≤ (C * r^n) / (1 - r) | begin
have := aux_has_sum_of_le_geometric hr hu,
convert dist_le_tsum_of_dist_le_of_tendsto _ hu ⟨_, this⟩ ha n,
simp only [pow_add, mul_left_comm C, mul_div_right_comm],
rw [mul_comm],
exact (this.mul_left _).tsum_eq.symm
end | lemma | dist_le_of_le_geometric_of_tendsto | analysis.specific_limits | src/analysis/specific_limits/basic.lean | [
"algebra.geom_sum",
"order.filter.archimedean",
"order.iterate",
"topology.instances.ennreal",
"topology.algebra.algebra"
] | [
"aux_has_sum_of_le_geometric",
"dist_le_tsum_of_dist_le_of_tendsto",
"mul_comm",
"mul_div_right_comm",
"mul_left_comm",
"pow_add"
] | If `dist (f n) (f (n+1))` is bounded by `C * r^n`, `r < 1`, then the distance from
`f 0` to the limit of `f` is bounded above by `C / (1 - r)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cauchy_seq_of_le_geometric_two : cauchy_seq f | cauchy_seq_of_dist_le_of_summable _ hu₂ $ ⟨_, has_sum_geometric_two' C⟩ | lemma | cauchy_seq_of_le_geometric_two | analysis.specific_limits | src/analysis/specific_limits/basic.lean | [
"algebra.geom_sum",
"order.filter.archimedean",
"order.iterate",
"topology.instances.ennreal",
"topology.algebra.algebra"
] | [
"cauchy_seq",
"cauchy_seq_of_dist_le_of_summable",
"has_sum_geometric_two'"
] | If `dist (f n) (f (n+1))` is bounded by `(C / 2) / 2^n`, then `f` is a Cauchy sequence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dist_le_of_le_geometric_two_of_tendsto₀ {a : α} (ha : tendsto f at_top (𝓝 a)) :
dist (f 0) a ≤ C | (tsum_geometric_two' C) ▸ dist_le_tsum_of_dist_le_of_tendsto₀ _ hu₂ (summable_geometric_two' C) ha | lemma | dist_le_of_le_geometric_two_of_tendsto₀ | analysis.specific_limits | src/analysis/specific_limits/basic.lean | [
"algebra.geom_sum",
"order.filter.archimedean",
"order.iterate",
"topology.instances.ennreal",
"topology.algebra.algebra"
] | [
"dist_le_tsum_of_dist_le_of_tendsto₀",
"summable_geometric_two'",
"tsum_geometric_two'"
] | If `dist (f n) (f (n+1))` is bounded by `(C / 2) / 2^n`, then the distance from
`f 0` to the limit of `f` is bounded above by `C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dist_le_of_le_geometric_two_of_tendsto {a : α} (ha : tendsto f at_top (𝓝 a)) (n : ℕ) :
dist (f n) a ≤ C / 2^n | begin
convert dist_le_tsum_of_dist_le_of_tendsto _ hu₂ (summable_geometric_two' C) ha n,
simp only [add_comm n, pow_add, ← div_div],
symmetry,
exact ((has_sum_geometric_two' C).div_const _).tsum_eq
end | lemma | dist_le_of_le_geometric_two_of_tendsto | analysis.specific_limits | src/analysis/specific_limits/basic.lean | [
"algebra.geom_sum",
"order.filter.archimedean",
"order.iterate",
"topology.instances.ennreal",
"topology.algebra.algebra"
] | [
"dist_le_tsum_of_dist_le_of_tendsto",
"div_div",
"has_sum_geometric_two'",
"pow_add",
"summable_geometric_two'"
] | If `dist (f n) (f (n+1))` is bounded by `(C / 2) / 2^n`, then the distance from
`f n` to the limit of `f` is bounded above by `C / 2^n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
summable_one_div_pow_of_le {m : ℝ} {f : ℕ → ℕ} (hm : 1 < m) (fi : ∀ i, i ≤ f i) :
summable (λ i, 1 / m ^ f i) | begin
refine summable_of_nonneg_of_le
(λ a, one_div_nonneg.mpr (pow_nonneg (zero_le_one.trans hm.le) _)) (λ a, _)
(summable_geometric_of_lt_1 (one_div_nonneg.mpr (zero_le_one.trans hm.le))
((one_div_lt (zero_lt_one.trans hm) zero_lt_one).mpr (one_div_one.le.trans_lt hm))),
rw [div_pow, one_pow],
ref... | lemma | summable_one_div_pow_of_le | analysis.specific_limits | src/analysis/specific_limits/basic.lean | [
"algebra.geom_sum",
"order.filter.archimedean",
"order.iterate",
"topology.instances.ennreal",
"topology.algebra.algebra"
] | [
"div_pow",
"one_div_le_one_div",
"one_div_lt",
"one_pow",
"pow_le_pow",
"pow_nonneg",
"pow_pos",
"summable",
"summable_geometric_of_lt_1",
"summable_of_nonneg_of_le",
"zero_lt_one"
] | A series whose terms are bounded by the terms of a converging geometric series converges. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pos_sum_of_encodable {ε : ℝ} (hε : 0 < ε)
(ι) [encodable ι] : {ε' : ι → ℝ // (∀ i, 0 < ε' i) ∧ ∃ c, has_sum ε' c ∧ c ≤ ε} | begin
let f := λ n, (ε / 2) / 2 ^ n,
have hf : has_sum f ε := has_sum_geometric_two' _,
have f0 : ∀ n, 0 < f n := λ n, div_pos (half_pos hε) (pow_pos zero_lt_two _),
refine ⟨f ∘ encodable.encode, λ i, f0 _, _⟩,
rcases hf.summable.comp_injective (@encodable.encode_injective ι _) with ⟨c, hg⟩,
refine ⟨c, hg, ... | def | pos_sum_of_encodable | analysis.specific_limits | src/analysis/specific_limits/basic.lean | [
"algebra.geom_sum",
"order.filter.archimedean",
"order.iterate",
"topology.instances.ennreal",
"topology.algebra.algebra"
] | [
"div_pos",
"encodable",
"encodable.encode_injective",
"half_pos",
"has_sum",
"has_sum_geometric_two'",
"has_sum_le_inj",
"le_rfl",
"pow_pos",
"zero_lt_two"
] | For any positive `ε`, define on an encodable type a positive sequence with sum less than `ε` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
set.countable.exists_pos_has_sum_le {ι : Type*} {s : set ι} (hs : s.countable)
{ε : ℝ} (hε : 0 < ε) :
∃ ε' : ι → ℝ, (∀ i, 0 < ε' i) ∧ ∃ c, has_sum (λ i : s, ε' i) c ∧ c ≤ ε | begin
haveI := hs.to_encodable,
rcases pos_sum_of_encodable hε s with ⟨f, hf0, ⟨c, hfc, hcε⟩⟩,
refine ⟨λ i, if h : i ∈ s then f ⟨i, h⟩ else 1, λ i, _, ⟨c, _, hcε⟩⟩,
{ split_ifs, exacts [hf0 _, zero_lt_one] },
{ simpa only [subtype.coe_prop, dif_pos, subtype.coe_eta] }
end | lemma | set.countable.exists_pos_has_sum_le | analysis.specific_limits | src/analysis/specific_limits/basic.lean | [
"algebra.geom_sum",
"order.filter.archimedean",
"order.iterate",
"topology.instances.ennreal",
"topology.algebra.algebra"
] | [
"has_sum",
"pos_sum_of_encodable",
"subtype.coe_eta",
"subtype.coe_prop",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set.countable.exists_pos_forall_sum_le {ι : Type*} {s : set ι} (hs : s.countable)
{ε : ℝ} (hε : 0 < ε) :
∃ ε' : ι → ℝ, (∀ i, 0 < ε' i) ∧ ∀ t : finset ι, ↑t ⊆ s → ∑ i in t, ε' i ≤ ε | begin
rcases hs.exists_pos_has_sum_le hε with ⟨ε', hpos, c, hε'c, hcε⟩,
refine ⟨ε', hpos, λ t ht, _⟩,
rw [← sum_subtype_of_mem _ ht],
refine (sum_le_has_sum _ _ hε'c).trans hcε,
exact λ _ _, (hpos _).le
end | lemma | set.countable.exists_pos_forall_sum_le | analysis.specific_limits | src/analysis/specific_limits/basic.lean | [
"algebra.geom_sum",
"order.filter.archimedean",
"order.iterate",
"topology.instances.ennreal",
"topology.algebra.algebra"
] | [
"finset",
"sum_le_has_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_pos_sum_of_countable {ε : ℝ≥0} (hε : ε ≠ 0) (ι) [countable ι] :
∃ ε' : ι → ℝ≥0, (∀ i, 0 < ε' i) ∧ ∃c, has_sum ε' c ∧ c < ε | begin
casesI nonempty_encodable ι,
obtain ⟨a, a0, aε⟩ := exists_between (pos_iff_ne_zero.2 hε),
obtain ⟨ε', hε', c, hc, hcε⟩ := pos_sum_of_encodable a0 ι,
exact ⟨λ i, ⟨ε' i, (hε' i).le⟩, λ i, nnreal.coe_lt_coe.1 $ hε' i, ⟨c, has_sum_le (λ i, (hε' i).le)
has_sum_zero hc⟩, nnreal.has_sum_coe.1 hc, aε.trans_le... | theorem | nnreal.exists_pos_sum_of_countable | analysis.specific_limits | src/analysis/specific_limits/basic.lean | [
"algebra.geom_sum",
"order.filter.archimedean",
"order.iterate",
"topology.instances.ennreal",
"topology.algebra.algebra"
] | [
"countable",
"exists_between",
"has_sum",
"has_sum_le",
"has_sum_zero",
"nonempty_encodable",
"pos_sum_of_encodable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_pos_sum_of_countable {ε : ℝ≥0∞} (hε : ε ≠ 0) (ι) [countable ι] :
∃ ε' : ι → ℝ≥0, (∀ i, 0 < ε' i) ∧ ∑' i, (ε' i : ℝ≥0∞) < ε | begin
rcases exists_between (pos_iff_ne_zero.2 hε) with ⟨r, h0r, hrε⟩,
rcases lt_iff_exists_coe.1 hrε with ⟨x, rfl, hx⟩,
rcases nnreal.exists_pos_sum_of_countable (coe_pos.1 h0r).ne' ι with ⟨ε', hp, c, hc, hcr⟩,
exact ⟨ε', hp, (ennreal.tsum_coe_eq hc).symm ▸ lt_trans (coe_lt_coe.2 hcr) hrε⟩
end | theorem | ennreal.exists_pos_sum_of_countable | analysis.specific_limits | src/analysis/specific_limits/basic.lean | [
"algebra.geom_sum",
"order.filter.archimedean",
"order.iterate",
"topology.instances.ennreal",
"topology.algebra.algebra"
] | [
"countable",
"ennreal.tsum_coe_eq",
"exists_between",
"nnreal.exists_pos_sum_of_countable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_pos_sum_of_countable' {ε : ℝ≥0∞} (hε : ε ≠ 0) (ι) [countable ι] :
∃ ε' : ι → ℝ≥0∞, (∀ i, 0 < ε' i) ∧ (∑' i, ε' i) < ε | let ⟨δ, δpos, hδ⟩ := exists_pos_sum_of_countable hε ι in
⟨λ i, δ i, λ i, ennreal.coe_pos.2 (δpos i), hδ⟩ | theorem | ennreal.exists_pos_sum_of_countable' | analysis.specific_limits | src/analysis/specific_limits/basic.lean | [
"algebra.geom_sum",
"order.filter.archimedean",
"order.iterate",
"topology.instances.ennreal",
"topology.algebra.algebra"
] | [
"countable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_pos_tsum_mul_lt_of_countable {ε : ℝ≥0∞} (hε : ε ≠ 0) {ι} [countable ι]
(w : ι → ℝ≥0∞) (hw : ∀ i, w i ≠ ∞) :
∃ δ : ι → ℝ≥0, (∀ i, 0 < δ i) ∧ ∑' i, (w i * δ i : ℝ≥0∞) < ε | begin
lift w to ι → ℝ≥0 using hw,
rcases exists_pos_sum_of_countable hε ι with ⟨δ', Hpos, Hsum⟩,
have : ∀ i, 0 < max 1 (w i), from λ i, zero_lt_one.trans_le (le_max_left _ _),
refine ⟨λ i, δ' i / max 1 (w i), λ i, div_pos (Hpos _) (this i), _⟩,
refine lt_of_le_of_lt (ennreal.tsum_le_tsum $ λ i, _) Hsum,
rw ... | theorem | ennreal.exists_pos_tsum_mul_lt_of_countable | analysis.specific_limits | src/analysis/specific_limits/basic.lean | [
"algebra.geom_sum",
"order.filter.archimedean",
"order.iterate",
"topology.instances.ennreal",
"topology.algebra.algebra"
] | [
"countable",
"div_pos",
"ennreal.tsum_le_tsum",
"lift",
"mul_le_mul_left'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
factorial_tendsto_at_top : tendsto nat.factorial at_top at_top | tendsto_at_top_at_top_of_monotone nat.monotone_factorial (λ n, ⟨n, n.self_le_factorial⟩) | lemma | factorial_tendsto_at_top | analysis.specific_limits | src/analysis/specific_limits/basic.lean | [
"algebra.geom_sum",
"order.filter.archimedean",
"order.iterate",
"topology.instances.ennreal",
"topology.algebra.algebra"
] | [
"nat.factorial",
"nat.monotone_factorial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_factorial_div_pow_self_at_top : tendsto (λ n, n! / n^n : ℕ → ℝ) at_top (𝓝 0) | tendsto_of_tendsto_of_tendsto_of_le_of_le'
tendsto_const_nhds
(tendsto_const_div_at_top_nhds_0_nat 1)
(eventually_of_forall $ λ n, div_nonneg (by exact_mod_cast n.factorial_pos.le)
(pow_nonneg (by exact_mod_cast n.zero_le) _))
begin
refine (eventually_gt_at_top 0).mono (λ n hn, _),
rcases nat.exists... | lemma | tendsto_factorial_div_pow_self_at_top | analysis.specific_limits | src/analysis/specific_limits/basic.lean | [
"algebra.geom_sum",
"order.filter.archimedean",
"order.iterate",
"topology.instances.ennreal",
"topology.algebra.algebra"
] | [
"div_eq_mul_inv",
"div_le_one",
"div_nonneg",
"finset.mem_range",
"finset.prod_range_succ'",
"inv_eq_one_div",
"mul_le_of_le_one_left",
"nat.cast_add",
"nat.cast_one",
"nat.cast_succ",
"one_mul",
"pow_nonneg",
"tendsto_const_div_at_top_nhds_0_nat",
"tendsto_const_nhds",
"tendsto_of_tends... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_nat_floor_at_top {α : Type*} [linear_ordered_semiring α] [floor_semiring α] :
tendsto (λ (x : α), ⌊x⌋₊) at_top at_top | nat.floor_mono.tendsto_at_top_at_top (λ x, ⟨max 0 (x + 1), by simp [nat.le_floor_iff]⟩) | lemma | tendsto_nat_floor_at_top | analysis.specific_limits | src/analysis/specific_limits/basic.lean | [
"algebra.geom_sum",
"order.filter.archimedean",
"order.iterate",
"topology.instances.ennreal",
"topology.algebra.algebra"
] | [
"floor_semiring",
"linear_ordered_semiring",
"nat.le_floor_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_nat_floor_mul_div_at_top {a : R} (ha : 0 ≤ a) :
tendsto (λ x, (⌊a * x⌋₊ : R) / x) at_top (𝓝 a) | begin
have A : tendsto (λ (x : R), a - x⁻¹) at_top (𝓝 (a - 0)) :=
tendsto_const_nhds.sub tendsto_inv_at_top_zero,
rw sub_zero at A,
apply tendsto_of_tendsto_of_tendsto_of_le_of_le' A tendsto_const_nhds,
{ refine eventually_at_top.2 ⟨1, λ x hx, _⟩,
simp only [le_div_iff (zero_lt_one.trans_le hx), sub_mu... | lemma | tendsto_nat_floor_mul_div_at_top | analysis.specific_limits | src/analysis/specific_limits/basic.lean | [
"algebra.geom_sum",
"order.filter.archimedean",
"order.iterate",
"topology.instances.ennreal",
"topology.algebra.algebra"
] | [
"div_le_iff",
"inv_mul_cancel",
"le_div_iff",
"nat.floor_le",
"nat.lt_floor_add_one",
"tendsto_const_nhds",
"tendsto_inv_at_top_zero",
"tendsto_of_tendsto_of_tendsto_of_le_of_le'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_nat_floor_div_at_top :
tendsto (λ x, (⌊x⌋₊ : R) / x) at_top (𝓝 1) | by simpa using tendsto_nat_floor_mul_div_at_top (zero_le_one' R) | lemma | tendsto_nat_floor_div_at_top | analysis.specific_limits | src/analysis/specific_limits/basic.lean | [
"algebra.geom_sum",
"order.filter.archimedean",
"order.iterate",
"topology.instances.ennreal",
"topology.algebra.algebra"
] | [
"tendsto_nat_floor_mul_div_at_top",
"zero_le_one'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_nat_ceil_mul_div_at_top {a : R} (ha : 0 ≤ a) :
tendsto (λ x, (⌈a * x⌉₊ : R) / x) at_top (𝓝 a) | begin
have A : tendsto (λ (x : R), a + x⁻¹) at_top (𝓝 (a + 0)) :=
tendsto_const_nhds.add tendsto_inv_at_top_zero,
rw add_zero at A,
apply tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds A,
{ refine eventually_at_top.2 ⟨1, λ x hx, _⟩,
rw le_div_iff (zero_lt_one.trans_le hx),
exact nat.... | lemma | tendsto_nat_ceil_mul_div_at_top | analysis.specific_limits | src/analysis/specific_limits/basic.lean | [
"algebra.geom_sum",
"order.filter.archimedean",
"order.iterate",
"topology.instances.ennreal",
"topology.algebra.algebra"
] | [
"div_le_iff",
"inv_mul_cancel",
"le_div_iff",
"nat.ceil_lt_add_one",
"nat.le_ceil",
"tendsto_const_nhds",
"tendsto_inv_at_top_zero",
"tendsto_of_tendsto_of_tendsto_of_le_of_le'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_nat_ceil_div_at_top :
tendsto (λ x, (⌈x⌉₊ : R) / x) at_top (𝓝 1) | by simpa using tendsto_nat_ceil_mul_div_at_top (zero_le_one' R) | lemma | tendsto_nat_ceil_div_at_top | analysis.specific_limits | src/analysis/specific_limits/basic.lean | [
"algebra.geom_sum",
"order.filter.archimedean",
"order.iterate",
"topology.instances.ennreal",
"topology.algebra.algebra"
] | [
"tendsto_nat_ceil_mul_div_at_top",
"zero_le_one'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (l : ℝ) (hmono : monotone u)
(hlim : ∀ (a : ℝ), 1 < a → ∃ c : ℕ → ℕ, (∀ᶠ n in at_top, (c (n+1) : ℝ) ≤ a * c n) ∧
tendsto c at_top at_top ∧ tendsto (λ n, u (c n) / (c n)) at_top (𝓝 l)) :
tendsto (λ n, u n / n) at_top (𝓝 l) | begin
/- To check the result up to some `ε > 0`, we use a sequence `c` for which the ratio
`c (N+1) / c N` is bounded by `1 + ε`. Sandwiching a given `n` between two consecutive values of
`c`, say `c N` and `c (N+1)`, one can then bound `u n / n` from above by `u (c N) / c (N - 1)`
and from below by `u (c (N - ... | lemma | tendsto_div_of_monotone_of_exists_subseq_tendsto_div | analysis.specific_limits | src/analysis/specific_limits/floor_pow.lean | [
"analysis.specific_limits.basic",
"analysis.special_functions.pow.real"
] | [
"abs_one",
"asymptotics.is_o_iff",
"asymptotics.is_o_one_iff",
"ctop",
"div_eq_inv_mul",
"finset.range",
"inv_mul_cancel",
"le_abs_self",
"le_of_tendsto_of_tendsto'",
"le_rfl",
"monotone",
"mul_le_mul_of_nonneg_left",
"mul_le_mul_of_nonneg_right",
"mul_one",
"nat.cast_eq_zero",
"nat.ca... | If a monotone sequence `u` is such that `u n / n` tends to a limit `l` along subsequences with
exponential growth rate arbitrarily close to `1`, then `u n / n` tends to `l`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_div_of_monotone_of_tendsto_div_floor_pow
(u : ℕ → ℝ) (l : ℝ) (hmono : monotone u)
(c : ℕ → ℝ) (cone : ∀ k, 1 < c k) (clim : tendsto c at_top (𝓝 1))
(hc : ∀ k, tendsto (λ (n : ℕ), u (⌊(c k) ^ n⌋₊) / ⌊(c k)^n⌋₊) at_top (𝓝 l)) :
tendsto (λ n, u n / n) at_top (𝓝 l) | begin
apply tendsto_div_of_monotone_of_exists_subseq_tendsto_div u l hmono,
assume a ha,
obtain ⟨k, hk⟩ : ∃ k, c k < a := ((tendsto_order.1 clim).2 a ha).exists,
refine ⟨λ n, ⌊(c k)^n⌋₊, _,
tendsto_nat_floor_at_top.comp (tendsto_pow_at_top_at_top_of_one_lt (cone k)), hc k⟩,
have H : ∀ (n : ℕ), (0 : ℝ) < ⌊... | lemma | tendsto_div_of_monotone_of_tendsto_div_floor_pow | analysis.specific_limits | src/analysis/specific_limits/floor_pow.lean | [
"analysis.specific_limits.basic",
"analysis.special_functions.pow.real"
] | [
"div_le_iff",
"div_one",
"monotone",
"nat.one_le_cast",
"nat.one_le_floor_iff",
"one_le_pow_of_one_le",
"one_mul",
"one_ne_zero",
"tactic.field_simp.ne_zero",
"tendsto_const_nhds",
"tendsto_div_of_monotone_of_exists_subseq_tendsto_div",
"tendsto_pow_at_top_at_top_of_one_lt"
] | If a monotone sequence `u` is such that `u ⌊c^n⌋₊ / ⌊c^n⌋₊` converges to a limit `l` for all
`c > 1`, then `u n / n` tends to `l`. It is even enough to have the assumption for a sequence of
`c`s converging to `1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sum_div_pow_sq_le_div_sq (N : ℕ) {j : ℝ} (hj : 0 < j) {c : ℝ} (hc : 1 < c) :
∑ i in (range N).filter (λ i, j < c ^ i), 1 / (c ^ i) ^ 2 ≤ (c^3 * (c - 1) ⁻¹) / j ^ 2 | begin
have cpos : 0 < c := zero_lt_one.trans hc,
have A : 0 < (c⁻¹) ^ 2 := sq_pos_of_pos (inv_pos.2 cpos),
have B : c^2 * (1 - c⁻¹ ^ 2) ⁻¹ ≤ c^3 * (c - 1) ⁻¹,
{ rw [← div_eq_mul_inv, ← div_eq_mul_inv, div_le_div_iff _ (sub_pos.2 hc)], swap,
{ exact sub_pos.2 (pow_lt_one (inv_nonneg.2 cpos.le) (inv_lt_one hc... | lemma | sum_div_pow_sq_le_div_sq | analysis.specific_limits | src/analysis/specific_limits/floor_pow.lean | [
"analysis.specific_limits.basic",
"analysis.special_functions.pow.real"
] | [
"div_eq_mul_inv",
"div_le_div",
"div_le_div_iff",
"div_lt_iff",
"div_nonneg",
"filter",
"geom_sum_Ico_le_of_lt_one",
"inv_le_one",
"inv_lt_one",
"inv_pow",
"le_rfl",
"mul_assoc",
"mul_comm",
"mul_inv_cancel",
"mul_neg",
"mul_one",
"nat.cast_bit0",
"nat.cast_one",
"nat.floor_le_of... | The sum of `1/(c^i)^2` above a threshold `j` is comparable to `1/j^2`, up to a multiplicative
constant. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_pow_le_nat_floor_pow {c : ℝ} (hc : 1 < c) (i : ℕ) :
(1 - c⁻¹) * c ^ i ≤ ⌊c ^ i⌋₊ | begin
have cpos : 0 < c := zero_lt_one.trans hc,
rcases nat.eq_zero_or_pos i with rfl|hi,
{ simp only [pow_zero, nat.floor_one, nat.cast_one, mul_one, sub_le_self_iff, inv_nonneg,
cpos.le] },
have hident : 1 ≤ i := hi,
calc (1 - c⁻¹) * c ^ i
= c ^ i - c ^ i * c ⁻¹ : by ring
... ≤ c ^ i - 1 :
... | lemma | mul_pow_le_nat_floor_pow | analysis.specific_limits | src/analysis/specific_limits/floor_pow.lean | [
"analysis.specific_limits.basic",
"analysis.special_functions.pow.real"
] | [
"inv_nonneg",
"mul_one",
"nat.cast_one",
"nat.floor_one",
"nat.sub_one_lt_floor",
"one_le_div",
"pow_le_pow",
"pow_one",
"pow_zero",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_div_nat_floor_pow_sq_le_div_sq (N : ℕ) {j : ℝ} (hj : 0 < j) {c : ℝ} (hc : 1 < c) :
∑ i in (range N).filter (λ i, j < ⌊c ^ i⌋₊), (1 : ℝ) / ⌊c ^ i⌋₊ ^ 2
≤ (c ^ 5 * (c - 1) ⁻¹ ^ 3) / j ^ 2 | begin
have cpos : 0 < c := zero_lt_one.trans hc,
have A : 0 < 1 - c⁻¹ := sub_pos.2 (inv_lt_one hc),
calc
∑ i in (range N).filter (λ i, j < ⌊c ^ i⌋₊), (1 : ℝ) / ⌊c ^ i⌋₊ ^ 2
≤ ∑ i in (range N).filter (λ i, j < c ^ i), (1 : ℝ) / ⌊c ^ i⌋₊ ^ 2 :
begin
apply sum_le_sum_of_subset_of_nonneg,
{ assume i... | lemma | sum_div_nat_floor_pow_sq_le_div_sq | analysis.specific_limits | src/analysis/specific_limits/floor_pow.lean | [
"analysis.specific_limits.basic",
"analysis.special_functions.pow.real"
] | [
"div_eq_inv_mul",
"div_le_div_iff",
"div_nonneg",
"filter",
"inv_lt_one",
"le_div_iff",
"mul_comm",
"mul_div_assoc'",
"mul_le_mul_of_nonneg_left",
"mul_one",
"mul_pow",
"mul_pow_le_nat_floor_pow",
"nat.cast_one",
"nat.floor_le",
"nat.le_floor",
"nat.one_le_cast",
"one_le_pow_of_one_l... | The sum of `1/⌊c^i⌋₊^2` above a threshold `j` is comparable to `1/j^2`, up to a multiplicative
constant. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_norm_at_top_at_top : tendsto (norm : ℝ → ℝ) at_top at_top | tendsto_abs_at_top_at_top | lemma | tendsto_norm_at_top_at_top | analysis.specific_limits | src/analysis/specific_limits/normed.lean | [
"algebra.order.field.basic",
"analysis.asymptotics.asymptotics",
"analysis.specific_limits.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_of_absolute_convergence_real {f : ℕ → ℝ} :
(∃r, tendsto (λn, (∑ i in range n, |f i|)) at_top (𝓝 r)) → summable f | | ⟨r, hr⟩ :=
begin
refine summable_of_summable_norm ⟨r, (has_sum_iff_tendsto_nat_of_nonneg _ _).2 _⟩,
exact assume i, norm_nonneg _,
simpa only using hr
end | lemma | summable_of_absolute_convergence_real | analysis.specific_limits | src/analysis/specific_limits/normed.lean | [
"algebra.order.field.basic",
"analysis.asymptotics.asymptotics",
"analysis.specific_limits.basic"
] | [
"has_sum_iff_tendsto_nat_of_nonneg",
"summable",
"summable_of_summable_norm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_norm_zero' {𝕜 : Type*} [normed_add_comm_group 𝕜] :
tendsto (norm : 𝕜 → ℝ) (𝓝[≠] 0) (𝓝[>] 0) | tendsto_norm_zero.inf $ tendsto_principal_principal.2 $ λ x hx, norm_pos_iff.2 hx | lemma | tendsto_norm_zero' | analysis.specific_limits | src/analysis/specific_limits/normed.lean | [
"algebra.order.field.basic",
"analysis.asymptotics.asymptotics",
"analysis.specific_limits.basic"
] | [
"normed_add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_norm_inverse_nhds_within_0_at_top {𝕜 : Type*} [normed_field 𝕜] :
tendsto (λ x:𝕜, ‖x⁻¹‖) (𝓝[≠] 0) at_top | (tendsto_inv_zero_at_top.comp tendsto_norm_zero').congr $ λ x, (norm_inv x).symm | lemma | normed_field.tendsto_norm_inverse_nhds_within_0_at_top | analysis.specific_limits | src/analysis/specific_limits/normed.lean | [
"algebra.order.field.basic",
"analysis.asymptotics.asymptotics",
"analysis.specific_limits.basic"
] | [
"norm_inv",
"normed_field",
"tendsto_norm_zero'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_norm_zpow_nhds_within_0_at_top {𝕜 : Type*} [normed_field 𝕜] {m : ℤ}
(hm : m < 0) :
tendsto (λ x : 𝕜, ‖x ^ m‖) (𝓝[≠] 0) at_top | begin
rcases neg_surjective m with ⟨m, rfl⟩,
rw neg_lt_zero at hm, lift m to ℕ using hm.le, rw int.coe_nat_pos at hm,
simp only [norm_pow, zpow_neg, zpow_coe_nat, ← inv_pow],
exact (tendsto_pow_at_top hm.ne').comp normed_field.tendsto_norm_inverse_nhds_within_0_at_top
end | lemma | normed_field.tendsto_norm_zpow_nhds_within_0_at_top | analysis.specific_limits | src/analysis/specific_limits/normed.lean | [
"algebra.order.field.basic",
"analysis.asymptotics.asymptotics",
"analysis.specific_limits.basic"
] | [
"int.coe_nat_pos",
"inv_pow",
"lift",
"norm_pow",
"normed_field",
"normed_field.tendsto_norm_inverse_nhds_within_0_at_top",
"zpow_coe_nat",
"zpow_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_zero_smul_of_tendsto_zero_of_bounded {ι 𝕜 𝔸 : Type*} [normed_field 𝕜]
[normed_add_comm_group 𝔸] [normed_space 𝕜 𝔸] {l : filter ι} {ε : ι → 𝕜} {f : ι → 𝔸}
(hε : tendsto ε l (𝓝 0)) (hf : filter.is_bounded_under (≤) l (norm ∘ f)) :
tendsto (ε • f) l (𝓝 0) | begin
rw ← is_o_one_iff 𝕜 at hε ⊢,
simpa using is_o.smul_is_O hε (hf.is_O_const (one_ne_zero : (1 : 𝕜) ≠ 0))
end | lemma | normed_field.tendsto_zero_smul_of_tendsto_zero_of_bounded | analysis.specific_limits | src/analysis/specific_limits/normed.lean | [
"algebra.order.field.basic",
"analysis.asymptotics.asymptotics",
"analysis.specific_limits.basic"
] | [
"filter",
"filter.is_bounded_under",
"normed_add_comm_group",
"normed_field",
"normed_space",
"one_ne_zero"
] | The (scalar) product of a sequence that tends to zero with a bounded one also tends to zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_at_zpow {𝕜 : Type*} [nontrivially_normed_field 𝕜] {m : ℤ} {x : 𝕜} :
continuous_at (λ x, x ^ m) x ↔ x ≠ 0 ∨ 0 ≤ m | begin
refine ⟨_, continuous_at_zpow₀ _ _⟩,
contrapose!, rintro ⟨rfl, hm⟩ hc,
exact not_tendsto_at_top_of_tendsto_nhds (hc.tendsto.mono_left nhds_within_le_nhds).norm
(tendsto_norm_zpow_nhds_within_0_at_top hm)
end | lemma | normed_field.continuous_at_zpow | analysis.specific_limits | src/analysis/specific_limits/normed.lean | [
"algebra.order.field.basic",
"analysis.asymptotics.asymptotics",
"analysis.specific_limits.basic"
] | [
"continuous_at",
"continuous_at_zpow",
"continuous_at_zpow₀",
"nhds_within_le_nhds",
"nontrivially_normed_field",
"not_tendsto_at_top_of_tendsto_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_inv {𝕜 : Type*} [nontrivially_normed_field 𝕜] {x : 𝕜} :
continuous_at has_inv.inv x ↔ x ≠ 0 | by simpa [(zero_lt_one' ℤ).not_le] using @continuous_at_zpow _ _ (-1) x | lemma | normed_field.continuous_at_inv | analysis.specific_limits | src/analysis/specific_limits/normed.lean | [
"algebra.order.field.basic",
"analysis.asymptotics.asymptotics",
"analysis.specific_limits.basic"
] | [
"continuous_at",
"continuous_at_inv",
"continuous_at_zpow",
"nontrivially_normed_field",
"zero_lt_one'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o_pow_pow_of_lt_left {r₁ r₂ : ℝ} (h₁ : 0 ≤ r₁) (h₂ : r₁ < r₂) :
(λ n : ℕ, r₁ ^ n) =o[at_top] (λ n, r₂ ^ n) | have H : 0 < r₂ := h₁.trans_lt h₂,
is_o_of_tendsto (λ n hn, false.elim $ H.ne' $ pow_eq_zero hn) $
(tendsto_pow_at_top_nhds_0_of_lt_1 (div_nonneg h₁ (h₁.trans h₂.le)) ((div_lt_one H).2 h₂)).congr
(λ n, div_pow _ _ _) | lemma | is_o_pow_pow_of_lt_left | analysis.specific_limits | src/analysis/specific_limits/normed.lean | [
"algebra.order.field.basic",
"analysis.asymptotics.asymptotics",
"analysis.specific_limits.basic"
] | [
"div_lt_one",
"div_nonneg",
"div_pow",
"pow_eq_zero",
"tendsto_pow_at_top_nhds_0_of_lt_1"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_pow_pow_of_le_left {r₁ r₂ : ℝ} (h₁ : 0 ≤ r₁) (h₂ : r₁ ≤ r₂) :
(λ n : ℕ, r₁ ^ n) =O[at_top] (λ n, r₂ ^ n) | h₂.eq_or_lt.elim (λ h, h ▸ is_O_refl _ _) (λ h, (is_o_pow_pow_of_lt_left h₁ h).is_O) | lemma | is_O_pow_pow_of_le_left | analysis.specific_limits | src/analysis/specific_limits/normed.lean | [
"algebra.order.field.basic",
"analysis.asymptotics.asymptotics",
"analysis.specific_limits.basic"
] | [
"is_o_pow_pow_of_lt_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o_pow_pow_of_abs_lt_left {r₁ r₂ : ℝ} (h : |r₁| < |r₂|) :
(λ n : ℕ, r₁ ^ n) =o[at_top] (λ n, r₂ ^ n) | begin
refine (is_o.of_norm_left _).of_norm_right,
exact (is_o_pow_pow_of_lt_left (abs_nonneg r₁) h).congr (pow_abs r₁) (pow_abs r₂)
end | lemma | is_o_pow_pow_of_abs_lt_left | analysis.specific_limits | src/analysis/specific_limits/normed.lean | [
"algebra.order.field.basic",
"analysis.asymptotics.asymptotics",
"analysis.specific_limits.basic"
] | [
"abs_nonneg",
"is_o_pow_pow_of_lt_left",
"pow_abs"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tfae_exists_lt_is_o_pow (f : ℕ → ℝ) (R : ℝ) :
tfae [∃ a ∈ Ioo (-R) R, f =o[at_top] pow a,
∃ a ∈ Ioo 0 R, f =o[at_top] (pow a),
∃ a ∈ Ioo (-R) R, f =O[at_top] pow a,
∃ a ∈ Ioo 0 R, f =O[at_top] pow a,
∃ (a < R) C (h₀ : 0 < C ∨ 0 < R), ∀ n, |f n| ≤ C * a ^ n,
∃ (a ∈ Ioo 0 R) (C > 0), ∀ n, |f n| ≤ C ... | begin
have A : Ico 0 R ⊆ Ioo (-R) R,
from λ x hx, ⟨(neg_lt_zero.2 (hx.1.trans_lt hx.2)).trans_le hx.1, hx.2⟩,
have B : Ioo 0 R ⊆ Ioo (-R) R := subset.trans Ioo_subset_Ico_self A,
-- First we prove that 1-4 are equivalent using 2 → 3 → 4, 1 → 3, and 2 → 1
tfae_have : 1 → 3, from λ ⟨a, ha, H⟩, ⟨a, ha, H.is_O⟩... | lemma | tfae_exists_lt_is_o_pow | analysis.specific_limits | src/analysis/specific_limits/normed.lean | [
"algebra.order.field.basic",
"analysis.asymptotics.asymptotics",
"analysis.specific_limits.basic"
] | [
"abs_nonneg",
"abs_of_nonneg",
"abs_of_pos",
"abs_pow",
"exists_between",
"is_o_pow_pow_of_abs_lt_left",
"le_abs_self",
"mul_le_mul_of_nonneg_left",
"one_mul",
"pow_ne_zero",
"real.norm_eq_abs",
"sign_cases_of_C_mul_pow_nonneg",
"zero_lt_one"
] | Various statements equivalent to the fact that `f n` grows exponentially slower than `R ^ n`.
* 0: $f n = o(a ^ n)$ for some $-R < a < R$;
* 1: $f n = o(a ^ n)$ for some $0 < a < R$;
* 2: $f n = O(a ^ n)$ for some $-R < a < R$;
* 3: $f n = O(a ^ n)$ for some $0 < a < R$;
* 4: there exist `a < R` and `C` such that one ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_o_pow_const_const_pow_of_one_lt {R : Type*} [normed_ring R] (k : ℕ) {r : ℝ} (hr : 1 < r) :
(λ n, n ^ k : ℕ → R) =o[at_top] (λ n, r ^ n) | begin
have : tendsto (λ x : ℝ, x ^ k) (𝓝[>] 1) (𝓝 1),
from ((continuous_id.pow k).tendsto' (1 : ℝ) 1 (one_pow _)).mono_left inf_le_left,
obtain ⟨r' : ℝ, hr' : r' ^ k < r, h1 : 1 < r'⟩ :=
((this.eventually (gt_mem_nhds hr)).and self_mem_nhds_within).exists,
have h0 : 0 ≤ r' := zero_le_one.trans h1.le,
... | lemma | is_o_pow_const_const_pow_of_one_lt | analysis.specific_limits | src/analysis/specific_limits/normed.lean | [
"algebra.order.field.basic",
"analysis.asymptotics.asymptotics",
"analysis.specific_limits.basic"
] | [
"abs_of_nonneg",
"div_eq_inv_mul",
"gt_mem_nhds",
"inf_le_left",
"is_o_pow_pow_of_lt_left",
"mul_comm",
"mul_le_mul_of_nonneg_right",
"mul_right_comm",
"normed_ring",
"one_pow",
"pow_mul",
"pow_nonneg",
"real.norm_eq_abs",
"self_mem_nhds_within"
] | For any natural `k` and a real `r > 1` we have `n ^ k = o(r ^ n)` as `n → ∞`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_o_coe_const_pow_of_one_lt {R : Type*} [normed_ring R] {r : ℝ} (hr : 1 < r) :
(coe : ℕ → R) =o[at_top] (λ n, r ^ n) | by simpa only [pow_one] using @is_o_pow_const_const_pow_of_one_lt R _ 1 _ hr | lemma | is_o_coe_const_pow_of_one_lt | analysis.specific_limits | src/analysis/specific_limits/normed.lean | [
"algebra.order.field.basic",
"analysis.asymptotics.asymptotics",
"analysis.specific_limits.basic"
] | [
"is_o_pow_const_const_pow_of_one_lt",
"normed_ring",
"pow_one"
] | For a real `r > 1` we have `n = o(r ^ n)` as `n → ∞`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_o_pow_const_mul_const_pow_const_pow_of_norm_lt {R : Type*} [normed_ring R] (k : ℕ)
{r₁ : R} {r₂ : ℝ} (h : ‖r₁‖ < r₂) :
(λ n, n ^ k * r₁ ^ n : ℕ → R) =o[at_top] (λ n, r₂ ^ n) | begin
by_cases h0 : r₁ = 0,
{ refine (is_o_zero _ _).congr' (mem_at_top_sets.2 $ ⟨1, λ n hn, _⟩) eventually_eq.rfl,
simp [zero_pow (zero_lt_one.trans_le hn), h0] },
rw [← ne.def, ← norm_pos_iff] at h0,
have A : (λ n, n ^ k : ℕ → R) =o[at_top] (λ n, (r₂ / ‖r₁‖) ^ n),
from is_o_pow_const_const_pow_of_one_... | lemma | is_o_pow_const_mul_const_pow_const_pow_of_norm_lt | analysis.specific_limits | src/analysis/specific_limits/normed.lean | [
"algebra.order.field.basic",
"analysis.asymptotics.asymptotics",
"analysis.specific_limits.basic"
] | [
"div_mul_cancel",
"eventually_norm_pow_le",
"is_o_pow_const_const_pow_of_one_lt",
"normed_ring",
"one_lt_div",
"pow_pos",
"zero_pow"
] | If `‖r₁‖ < r₂`, then for any naturak `k` we have `n ^ k r₁ ^ n = o (r₂ ^ n)` as `n → ∞`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_pow_const_div_const_pow_of_one_lt (k : ℕ) {r : ℝ} (hr : 1 < r) :
tendsto (λ n, n ^ k / r ^ n : ℕ → ℝ) at_top (𝓝 0) | (is_o_pow_const_const_pow_of_one_lt k hr).tendsto_div_nhds_zero | lemma | tendsto_pow_const_div_const_pow_of_one_lt | analysis.specific_limits | src/analysis/specific_limits/normed.lean | [
"algebra.order.field.basic",
"analysis.asymptotics.asymptotics",
"analysis.specific_limits.basic"
] | [
"is_o_pow_const_const_pow_of_one_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_pow_const_mul_const_pow_of_abs_lt_one (k : ℕ) {r : ℝ} (hr : |r| < 1) :
tendsto (λ n, n ^ k * r ^ n : ℕ → ℝ) at_top (𝓝 0) | begin
by_cases h0 : r = 0,
{ exact tendsto_const_nhds.congr'
(mem_at_top_sets.2 ⟨1, λ n hn, by simp [zero_lt_one.trans_le hn, h0]⟩) },
have hr' : 1 < (|r|)⁻¹, from one_lt_inv (abs_pos.2 h0) hr,
rw tendsto_zero_iff_norm_tendsto_zero,
simpa [div_eq_mul_inv] using tendsto_pow_const_div_const_pow_of_one_lt ... | lemma | tendsto_pow_const_mul_const_pow_of_abs_lt_one | analysis.specific_limits | src/analysis/specific_limits/normed.lean | [
"algebra.order.field.basic",
"analysis.asymptotics.asymptotics",
"analysis.specific_limits.basic"
] | [
"div_eq_mul_inv",
"one_lt_inv",
"tendsto_pow_const_div_const_pow_of_one_lt"
] | If `|r| < 1`, then `n ^ k r ^ n` tends to zero for any natural `k`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_pow_const_mul_const_pow_of_lt_one (k : ℕ) {r : ℝ} (hr : 0 ≤ r) (h'r : r < 1) :
tendsto (λ n, n ^ k * r ^ n : ℕ → ℝ) at_top (𝓝 0) | tendsto_pow_const_mul_const_pow_of_abs_lt_one k (abs_lt.2 ⟨neg_one_lt_zero.trans_le hr, h'r⟩) | lemma | tendsto_pow_const_mul_const_pow_of_lt_one | analysis.specific_limits | src/analysis/specific_limits/normed.lean | [
"algebra.order.field.basic",
"analysis.asymptotics.asymptotics",
"analysis.specific_limits.basic"
] | [
"tendsto_pow_const_mul_const_pow_of_abs_lt_one"
] | If `0 ≤ r < 1`, then `n ^ k r ^ n` tends to zero for any natural `k`.
This is a specialized version of `tendsto_pow_const_mul_const_pow_of_abs_lt_one`, singled out
for ease of application. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_self_mul_const_pow_of_abs_lt_one {r : ℝ} (hr : |r| < 1) :
tendsto (λ n, n * r ^ n : ℕ → ℝ) at_top (𝓝 0) | by simpa only [pow_one] using tendsto_pow_const_mul_const_pow_of_abs_lt_one 1 hr | lemma | tendsto_self_mul_const_pow_of_abs_lt_one | analysis.specific_limits | src/analysis/specific_limits/normed.lean | [
"algebra.order.field.basic",
"analysis.asymptotics.asymptotics",
"analysis.specific_limits.basic"
] | [
"pow_one",
"tendsto_pow_const_mul_const_pow_of_abs_lt_one"
] | If `|r| < 1`, then `n * r ^ n` tends to zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_self_mul_const_pow_of_lt_one {r : ℝ} (hr : 0 ≤ r) (h'r : r < 1) :
tendsto (λ n, n * r ^ n : ℕ → ℝ) at_top (𝓝 0) | by simpa only [pow_one] using tendsto_pow_const_mul_const_pow_of_lt_one 1 hr h'r | lemma | tendsto_self_mul_const_pow_of_lt_one | analysis.specific_limits | src/analysis/specific_limits/normed.lean | [
"algebra.order.field.basic",
"analysis.asymptotics.asymptotics",
"analysis.specific_limits.basic"
] | [
"pow_one",
"tendsto_pow_const_mul_const_pow_of_lt_one"
] | If `0 ≤ r < 1`, then `n * r ^ n` tends to zero. This is a specialized version of
`tendsto_self_mul_const_pow_of_abs_lt_one`, singled out for ease of application. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_pow_at_top_nhds_0_of_norm_lt_1 {R : Type*} [normed_ring R] {x : R}
(h : ‖x‖ < 1) : tendsto (λ (n : ℕ), x ^ n) at_top (𝓝 0) | begin
apply squeeze_zero_norm' (eventually_norm_pow_le x),
exact tendsto_pow_at_top_nhds_0_of_lt_1 (norm_nonneg _) h,
end | lemma | tendsto_pow_at_top_nhds_0_of_norm_lt_1 | analysis.specific_limits | src/analysis/specific_limits/normed.lean | [
"algebra.order.field.basic",
"analysis.asymptotics.asymptotics",
"analysis.specific_limits.basic"
] | [
"eventually_norm_pow_le",
"normed_ring",
"tendsto_pow_at_top_nhds_0_of_lt_1"
] | In a normed ring, the powers of an element x with `‖x‖ < 1` tend to zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_pow_at_top_nhds_0_of_abs_lt_1 {r : ℝ} (h : |r| < 1) :
tendsto (λn:ℕ, r^n) at_top (𝓝 0) | tendsto_pow_at_top_nhds_0_of_norm_lt_1 h | lemma | tendsto_pow_at_top_nhds_0_of_abs_lt_1 | analysis.specific_limits | src/analysis/specific_limits/normed.lean | [
"algebra.order.field.basic",
"analysis.asymptotics.asymptotics",
"analysis.specific_limits.basic"
] | [
"tendsto_pow_at_top_nhds_0_of_norm_lt_1"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_geometric_of_norm_lt_1 (h : ‖ξ‖ < 1) : has_sum (λn:ℕ, ξ ^ n) (1 - ξ)⁻¹ | begin
have xi_ne_one : ξ ≠ 1, by { contrapose! h, simp [h] },
have A : tendsto (λn, (ξ ^ n - 1) * (ξ - 1)⁻¹) at_top (𝓝 ((0 - 1) * (ξ - 1)⁻¹)),
from ((tendsto_pow_at_top_nhds_0_of_norm_lt_1 h).sub tendsto_const_nhds).mul tendsto_const_nhds,
rw [has_sum_iff_tendsto_nat_of_summable_norm],
{ simpa [geom_sum_eq... | lemma | has_sum_geometric_of_norm_lt_1 | analysis.specific_limits | src/analysis/specific_limits/normed.lean | [
"algebra.order.field.basic",
"analysis.asymptotics.asymptotics",
"analysis.specific_limits.basic"
] | [
"div_eq_mul_inv",
"geom_sum_eq",
"has_sum",
"has_sum_iff_tendsto_nat_of_summable_norm",
"neg_inv",
"norm_pow",
"summable_geometric_of_lt_1",
"tendsto_const_nhds",
"tendsto_pow_at_top_nhds_0_of_norm_lt_1"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_geometric_of_norm_lt_1 (h : ‖ξ‖ < 1) : summable (λn:ℕ, ξ ^ n) | ⟨_, has_sum_geometric_of_norm_lt_1 h⟩ | lemma | summable_geometric_of_norm_lt_1 | analysis.specific_limits | src/analysis/specific_limits/normed.lean | [
"algebra.order.field.basic",
"analysis.asymptotics.asymptotics",
"analysis.specific_limits.basic"
] | [
"has_sum_geometric_of_norm_lt_1",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_geometric_of_norm_lt_1 (h : ‖ξ‖ < 1) : ∑'n:ℕ, ξ ^ n = (1 - ξ)⁻¹ | (has_sum_geometric_of_norm_lt_1 h).tsum_eq | lemma | tsum_geometric_of_norm_lt_1 | analysis.specific_limits | src/analysis/specific_limits/normed.lean | [
"algebra.order.field.basic",
"analysis.asymptotics.asymptotics",
"analysis.specific_limits.basic"
] | [
"has_sum_geometric_of_norm_lt_1"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_geometric_of_abs_lt_1 {r : ℝ} (h : |r| < 1) : has_sum (λn:ℕ, r ^ n) (1 - r)⁻¹ | has_sum_geometric_of_norm_lt_1 h | lemma | has_sum_geometric_of_abs_lt_1 | analysis.specific_limits | src/analysis/specific_limits/normed.lean | [
"algebra.order.field.basic",
"analysis.asymptotics.asymptotics",
"analysis.specific_limits.basic"
] | [
"has_sum",
"has_sum_geometric_of_norm_lt_1"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_geometric_of_abs_lt_1 {r : ℝ} (h : |r| < 1) : summable (λn:ℕ, r ^ n) | summable_geometric_of_norm_lt_1 h | lemma | summable_geometric_of_abs_lt_1 | analysis.specific_limits | src/analysis/specific_limits/normed.lean | [
"algebra.order.field.basic",
"analysis.asymptotics.asymptotics",
"analysis.specific_limits.basic"
] | [
"summable",
"summable_geometric_of_norm_lt_1"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_geometric_of_abs_lt_1 {r : ℝ} (h : |r| < 1) : ∑'n:ℕ, r ^ n = (1 - r)⁻¹ | tsum_geometric_of_norm_lt_1 h | lemma | tsum_geometric_of_abs_lt_1 | analysis.specific_limits | src/analysis/specific_limits/normed.lean | [
"algebra.order.field.basic",
"analysis.asymptotics.asymptotics",
"analysis.specific_limits.basic"
] | [
"tsum_geometric_of_norm_lt_1"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_geometric_iff_norm_lt_1 : summable (λ n : ℕ, ξ ^ n) ↔ ‖ξ‖ < 1 | begin
refine ⟨λ h, _, summable_geometric_of_norm_lt_1⟩,
obtain ⟨k : ℕ, hk : dist (ξ ^ k) 0 < 1⟩ :=
(h.tendsto_cofinite_zero.eventually (ball_mem_nhds _ zero_lt_one)).exists,
simp only [norm_pow, dist_zero_right] at hk,
rw [← one_pow k] at hk,
exact lt_of_pow_lt_pow _ zero_le_one hk
end | lemma | summable_geometric_iff_norm_lt_1 | analysis.specific_limits | src/analysis/specific_limits/normed.lean | [
"algebra.order.field.basic",
"analysis.asymptotics.asymptotics",
"analysis.specific_limits.basic"
] | [
"lt_of_pow_lt_pow",
"norm_pow",
"one_pow",
"summable",
"zero_le_one",
"zero_lt_one"
] | A geometric series in a normed field is summable iff the norm of the common ratio is less than
one. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
summable_norm_pow_mul_geometric_of_norm_lt_1 {R : Type*} [normed_ring R]
(k : ℕ) {r : R} (hr : ‖r‖ < 1) : summable (λ n : ℕ, ‖(n ^ k * r ^ n : R)‖) | begin
rcases exists_between hr with ⟨r', hrr', h⟩,
exact summable_of_is_O_nat (summable_geometric_of_lt_1 ((norm_nonneg _).trans hrr'.le) h)
(is_o_pow_const_mul_const_pow_const_pow_of_norm_lt _ hrr').is_O.norm_left
end | lemma | summable_norm_pow_mul_geometric_of_norm_lt_1 | analysis.specific_limits | src/analysis/specific_limits/normed.lean | [
"algebra.order.field.basic",
"analysis.asymptotics.asymptotics",
"analysis.specific_limits.basic"
] | [
"exists_between",
"is_o_pow_const_mul_const_pow_const_pow_of_norm_lt",
"normed_ring",
"summable",
"summable_geometric_of_lt_1",
"summable_of_is_O_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_pow_mul_geometric_of_norm_lt_1 {R : Type*} [normed_ring R] [complete_space R]
(k : ℕ) {r : R} (hr : ‖r‖ < 1) : summable (λ n, n ^ k * r ^ n : ℕ → R) | summable_of_summable_norm $ summable_norm_pow_mul_geometric_of_norm_lt_1 _ hr | lemma | summable_pow_mul_geometric_of_norm_lt_1 | analysis.specific_limits | src/analysis/specific_limits/normed.lean | [
"algebra.order.field.basic",
"analysis.asymptotics.asymptotics",
"analysis.specific_limits.basic"
] | [
"complete_space",
"normed_ring",
"summable",
"summable_norm_pow_mul_geometric_of_norm_lt_1",
"summable_of_summable_norm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_coe_mul_geometric_of_norm_lt_1 {𝕜 : Type*} [normed_field 𝕜] [complete_space 𝕜]
{r : 𝕜} (hr : ‖r‖ < 1) : has_sum (λ n, n * r ^ n : ℕ → 𝕜) (r / (1 - r) ^ 2) | begin
have A : summable (λ n, n * r ^ n : ℕ → 𝕜),
by simpa using summable_pow_mul_geometric_of_norm_lt_1 1 hr,
have B : has_sum (pow r : ℕ → 𝕜) (1 - r)⁻¹, from has_sum_geometric_of_norm_lt_1 hr,
refine A.has_sum_iff.2 _,
have hr' : r ≠ 1, by { rintro rfl, simpa [lt_irrefl] using hr },
set s : 𝕜 := ∑' n... | lemma | has_sum_coe_mul_geometric_of_norm_lt_1 | analysis.specific_limits | src/analysis/specific_limits/normed.lean | [
"algebra.order.field.basic",
"analysis.asymptotics.asymptotics",
"analysis.specific_limits.basic"
] | [
"complete_space",
"div_div",
"div_eq_mul_inv",
"has_sum",
"has_sum_geometric_of_norm_lt_1",
"mul_div_cancel_left",
"mul_left_comm",
"normed_field",
"one_mul",
"pow_succ",
"summable",
"summable_pow_mul_geometric_of_norm_lt_1",
"tsum_add",
"tsum_eq_zero_add",
"tsum_mul_left"
] | If `‖r‖ < 1`, then `∑' n : ℕ, n * r ^ n = r / (1 - r) ^ 2`, `has_sum` version. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tsum_coe_mul_geometric_of_norm_lt_1 {𝕜 : Type*} [normed_field 𝕜] [complete_space 𝕜]
{r : 𝕜} (hr : ‖r‖ < 1) :
(∑' n : ℕ, n * r ^ n : 𝕜) = (r / (1 - r) ^ 2) | (has_sum_coe_mul_geometric_of_norm_lt_1 hr).tsum_eq | lemma | tsum_coe_mul_geometric_of_norm_lt_1 | analysis.specific_limits | src/analysis/specific_limits/normed.lean | [
"algebra.order.field.basic",
"analysis.asymptotics.asymptotics",
"analysis.specific_limits.basic"
] | [
"complete_space",
"has_sum_coe_mul_geometric_of_norm_lt_1",
"normed_field"
] | If `‖r‖ < 1`, then `∑' n : ℕ, n * r ^ n = r / (1 - r) ^ 2`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
seminormed_add_comm_group.cauchy_seq_of_le_geometric {C : ℝ} {r : ℝ} (hr : r < 1)
{u : ℕ → α} (h : ∀ n, ‖u n - u (n + 1)‖ ≤ C*r^n) : cauchy_seq u | cauchy_seq_of_le_geometric r C hr (by simpa [dist_eq_norm] using h) | lemma | seminormed_add_comm_group.cauchy_seq_of_le_geometric | analysis.specific_limits | src/analysis/specific_limits/normed.lean | [
"algebra.order.field.basic",
"analysis.asymptotics.asymptotics",
"analysis.specific_limits.basic"
] | [
"cauchy_seq",
"cauchy_seq_of_le_geometric"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_partial_sum_le_of_le_geometric (hf : ∀n, ‖f n‖ ≤ C * r^n) (n : ℕ) :
dist (∑ i in range n, f i) (∑ i in range (n+1), f i) ≤ C * r ^ n | begin
rw [sum_range_succ, dist_eq_norm, ← norm_neg, neg_sub, add_sub_cancel'],
exact hf n,
end | lemma | dist_partial_sum_le_of_le_geometric | analysis.specific_limits | src/analysis/specific_limits/normed.lean | [
"algebra.order.field.basic",
"analysis.asymptotics.asymptotics",
"analysis.specific_limits.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cauchy_seq_finset_of_geometric_bound (hr : r < 1) (hf : ∀n, ‖f n‖ ≤ C * r^n) :
cauchy_seq (λ s : finset (ℕ), ∑ x in s, f x) | cauchy_seq_finset_of_norm_bounded _
(aux_has_sum_of_le_geometric hr (dist_partial_sum_le_of_le_geometric hf)).summable hf | lemma | cauchy_seq_finset_of_geometric_bound | analysis.specific_limits | src/analysis/specific_limits/normed.lean | [
"algebra.order.field.basic",
"analysis.asymptotics.asymptotics",
"analysis.specific_limits.basic"
] | [
"aux_has_sum_of_le_geometric",
"cauchy_seq",
"cauchy_seq_finset_of_norm_bounded",
"dist_partial_sum_le_of_le_geometric",
"finset",
"summable"
] | If `‖f n‖ ≤ C * r ^ n` for all `n : ℕ` and some `r < 1`, then the partial sums of `f` form a
Cauchy sequence. This lemma does not assume `0 ≤ r` or `0 ≤ C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_sub_le_of_geometric_bound_of_has_sum (hr : r < 1) (hf : ∀n, ‖f n‖ ≤ C * r^n)
{a : α} (ha : has_sum f a) (n : ℕ) :
‖(∑ x in finset.range n, f x) - a‖ ≤ (C * r ^ n) / (1 - r) | begin
rw ← dist_eq_norm,
apply dist_le_of_le_geometric_of_tendsto r C hr (dist_partial_sum_le_of_le_geometric hf),
exact ha.tendsto_sum_nat
end | lemma | norm_sub_le_of_geometric_bound_of_has_sum | analysis.specific_limits | src/analysis/specific_limits/normed.lean | [
"algebra.order.field.basic",
"analysis.asymptotics.asymptotics",
"analysis.specific_limits.basic"
] | [
"dist_le_of_le_geometric_of_tendsto",
"dist_partial_sum_le_of_le_geometric",
"finset.range",
"has_sum"
] | If `‖f n‖ ≤ C * r ^ n` for all `n : ℕ` and some `r < 1`, then the partial sums of `f` are within
distance `C * r ^ n / (1 - r)` of the sum of the series. This lemma does not assume `0 ≤ r` or
`0 ≤ C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dist_partial_sum (u : ℕ → α) (n : ℕ) :
dist (∑ k in range (n + 1), u k) (∑ k in range n, u k) = ‖u n‖ | by simp [dist_eq_norm, sum_range_succ] | lemma | dist_partial_sum | analysis.specific_limits | src/analysis/specific_limits/normed.lean | [
"algebra.order.field.basic",
"analysis.asymptotics.asymptotics",
"analysis.specific_limits.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_partial_sum' (u : ℕ → α) (n : ℕ) :
dist (∑ k in range n, u k) (∑ k in range (n+1), u k) = ‖u n‖ | by simp [dist_eq_norm', sum_range_succ] | lemma | dist_partial_sum' | analysis.specific_limits | src/analysis/specific_limits/normed.lean | [
"algebra.order.field.basic",
"analysis.asymptotics.asymptotics",
"analysis.specific_limits.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cauchy_series_of_le_geometric {C : ℝ} {u : ℕ → α}
{r : ℝ} (hr : r < 1) (h : ∀ n, ‖u n‖ ≤ C*r^n) : cauchy_seq (λ n, ∑ k in range n, u k) | cauchy_seq_of_le_geometric r C hr (by simp [h]) | lemma | cauchy_series_of_le_geometric | analysis.specific_limits | src/analysis/specific_limits/normed.lean | [
"algebra.order.field.basic",
"analysis.asymptotics.asymptotics",
"analysis.specific_limits.basic"
] | [
"cauchy_seq",
"cauchy_seq_of_le_geometric"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normed_add_comm_group.cauchy_series_of_le_geometric' {C : ℝ} {u : ℕ → α} {r : ℝ} (hr : r < 1)
(h : ∀ n, ‖u n‖ ≤ C*r^n) : cauchy_seq (λ n, ∑ k in range (n + 1), u k) | (cauchy_series_of_le_geometric hr h).comp_tendsto $ tendsto_add_at_top_nat 1 | lemma | normed_add_comm_group.cauchy_series_of_le_geometric' | analysis.specific_limits | src/analysis/specific_limits/normed.lean | [
"algebra.order.field.basic",
"analysis.asymptotics.asymptotics",
"analysis.specific_limits.basic"
] | [
"cauchy_seq",
"cauchy_series_of_le_geometric"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normed_add_comm_group.cauchy_series_of_le_geometric'' {C : ℝ} {u : ℕ → α} {N : ℕ} {r : ℝ}
(hr₀ : 0 < r) (hr₁ : r < 1)
(h : ∀ n ≥ N, ‖u n‖ ≤ C*r^n) : cauchy_seq (λ n, ∑ k in range (n + 1), u k) | begin
set v : ℕ → α := λ n, if n < N then 0 else u n,
have hC : 0 ≤ C,
from (zero_le_mul_right $ pow_pos hr₀ N).mp ((norm_nonneg _).trans $ h N $ le_refl N),
have : ∀ n ≥ N, u n = v n,
{ intros n hn,
simp [v, hn, if_neg (not_lt.mpr hn)] },
refine cauchy_seq_sum_of_eventually_eq this (normed_add_comm_g... | lemma | normed_add_comm_group.cauchy_series_of_le_geometric'' | analysis.specific_limits | src/analysis/specific_limits/normed.lean | [
"algebra.order.field.basic",
"analysis.asymptotics.asymptotics",
"analysis.specific_limits.basic"
] | [
"cauchy_seq",
"normed_add_comm_group.cauchy_series_of_le_geometric'",
"pow_nonneg",
"pow_pos",
"zero_le_mul_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normed_ring.summable_geometric_of_norm_lt_1
(x : R) (h : ‖x‖ < 1) : summable (λ (n:ℕ), x ^ n) | begin
have h1 : summable (λ (n:ℕ), ‖x‖ ^ n) := summable_geometric_of_lt_1 (norm_nonneg _) h,
refine summable_of_norm_bounded_eventually _ h1 _,
rw nat.cofinite_eq_at_top,
exact eventually_norm_pow_le x,
end | lemma | normed_ring.summable_geometric_of_norm_lt_1 | analysis.specific_limits | src/analysis/specific_limits/normed.lean | [
"algebra.order.field.basic",
"analysis.asymptotics.asymptotics",
"analysis.specific_limits.basic"
] | [
"eventually_norm_pow_le",
"nat.cofinite_eq_at_top",
"summable",
"summable_geometric_of_lt_1",
"summable_of_norm_bounded_eventually"
] | A geometric series in a complete normed ring is summable.
Proved above (same name, different namespace) for not-necessarily-complete normed fields. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normed_ring.tsum_geometric_of_norm_lt_1
(x : R) (h : ‖x‖ < 1) : ‖∑' n:ℕ, x ^ n‖ ≤ ‖(1:R)‖ - 1 + (1 - ‖x‖)⁻¹ | begin
rw tsum_eq_zero_add (normed_ring.summable_geometric_of_norm_lt_1 x h),
simp only [pow_zero],
refine le_trans (norm_add_le _ _) _,
have : ‖∑' b : ℕ, (λ n, x ^ (n + 1)) b‖ ≤ (1 - ‖x‖)⁻¹ - 1,
{ refine tsum_of_norm_bounded _ (λ b, norm_pow_le' _ (nat.succ_pos b)),
convert (has_sum_nat_add_iff' 1).mpr (h... | lemma | normed_ring.tsum_geometric_of_norm_lt_1 | analysis.specific_limits | src/analysis/specific_limits/normed.lean | [
"algebra.order.field.basic",
"analysis.asymptotics.asymptotics",
"analysis.specific_limits.basic"
] | [
"has_sum_geometric_of_lt_1",
"has_sum_nat_add_iff'",
"norm_pow_le'",
"normed_ring.summable_geometric_of_norm_lt_1",
"pow_zero",
"tsum_eq_zero_add",
"tsum_of_norm_bounded"
] | Bound for the sum of a geometric series in a normed ring. This formula does not assume that the
normed ring satisfies the axiom `‖1‖ = 1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.