Theorem limsupbnd1 15191

Description: If a sequence is eventually at most 𝐴, then the limsup is also at most 𝐴. (The converse is only true if the less or equal is replaced by strictly less than; consider the sequence 1 / 𝑛 which is never less or equal to zero even though the limsup is.) (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by AV, 12-Sep-2020.)

Hypotheses

Ref	Expression
limsupbnd.1	⊢ (𝜑 → 𝐵 ⊆ ℝ)
limsupbnd.2	⊢ (𝜑 → 𝐹:𝐵⟶ℝ*)
limsupbnd.3	⊢ (𝜑 → 𝐴 ∈ ℝ*)
limsupbnd1.4	⊢ (𝜑 → ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴))

Assertion

Ref	Expression
limsupbnd1	⊢ (𝜑 → (lim sup‘𝐹) ≤ 𝐴)

Distinct variable groups:   𝑗,𝑘,𝐴   𝐵,𝑗,𝑘   𝑗,𝐹,𝑘   𝜑,𝑗,𝑘

Proof of Theorem limsupbnd1

Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		limsupbnd1.4	. 2 ⊢ (𝜑 → ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴))
2		limsupbnd.1	. . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ ℝ)
3	2	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℝ) → 𝐵 ⊆ ℝ)
4		limsupbnd.2	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐵⟶ℝ*)
5	4	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℝ) → 𝐹:𝐵⟶ℝ*)
6		simpr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℝ) → 𝑘 ∈ ℝ)
7		limsupbnd.3	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ*)
8	7	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℝ) → 𝐴 ∈ ℝ*)
9		eqid 2738	. . . . . 6 ⊢ (𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))
10	9	limsupgle 15186	. . . . 5 ⊢ (((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ*) ∧ 𝑘 ∈ ℝ ∧ 𝐴 ∈ ℝ*) → (((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑘) ≤ 𝐴 ↔ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴)))
11	3, 5, 6, 8, 10	syl211anc 1375	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℝ) → (((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑘) ≤ 𝐴 ↔ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴)))
12		reex 10962	. . . . . . . . . . . 12 ⊢ ℝ ∈ V
13	12	ssex 5245	. . . . . . . . . . 11 ⊢ (𝐵 ⊆ ℝ → 𝐵 ∈ V)
14	2, 13	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ V)
15		xrex 12727	. . . . . . . . . . 11 ⊢ ℝ* ∈ V
16	15	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ℝ* ∈ V)
17		fex2 7780	. . . . . . . . . 10 ⊢ ((𝐹:𝐵⟶ℝ* ∧ 𝐵 ∈ V ∧ ℝ* ∈ V) → 𝐹 ∈ V)
18	4, 14, 16, 17	syl3anc 1370	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ V)
19		limsupcl 15182	. . . . . . . . 9 ⊢ (𝐹 ∈ V → (lim sup‘𝐹) ∈ ℝ*)
20	18, 19	syl 17	. . . . . . . 8 ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ*)
21	20	xrleidd 12886	. . . . . . 7 ⊢ (𝜑 → (lim sup‘𝐹) ≤ (lim sup‘𝐹))
22	9	limsuple 15187	. . . . . . . 8 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ* ∧ (lim sup‘𝐹) ∈ ℝ*) → ((lim sup‘𝐹) ≤ (lim sup‘𝐹) ↔ ∀𝑘 ∈ ℝ (lim sup‘𝐹) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑘)))
23	2, 4, 20, 22	syl3anc 1370	. . . . . . 7 ⊢ (𝜑 → ((lim sup‘𝐹) ≤ (lim sup‘𝐹) ↔ ∀𝑘 ∈ ℝ (lim sup‘𝐹) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑘)))
24	21, 23	mpbid 231	. . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ ℝ (lim sup‘𝐹) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑘))
25	24	r19.21bi 3134	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℝ) → (lim sup‘𝐹) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑘))
26	20	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℝ) → (lim sup‘𝐹) ∈ ℝ*)
27	9	limsupgf 15184	. . . . . . . 8 ⊢ (𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < )):ℝ⟶ℝ*
28	27	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < )):ℝ⟶ℝ*)
29	28	ffvelrnda 6961	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℝ) → ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑘) ∈ ℝ*)
30		xrletr 12892	. . . . . 6 ⊢ (((lim sup‘𝐹) ∈ ℝ* ∧ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑘) ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → (((lim sup‘𝐹) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑘) ∧ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑘) ≤ 𝐴) → (lim sup‘𝐹) ≤ 𝐴))
31	26, 29, 8, 30	syl3anc 1370	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℝ) → (((lim sup‘𝐹) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑘) ∧ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑘) ≤ 𝐴) → (lim sup‘𝐹) ≤ 𝐴))
32	25, 31	mpand 692	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℝ) → (((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑘) ≤ 𝐴 → (lim sup‘𝐹) ≤ 𝐴))
33	11, 32	sylbird 259	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℝ) → (∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴) → (lim sup‘𝐹) ≤ 𝐴))
34	33	rexlimdva 3213	. 2 ⊢ (𝜑 → (∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴) → (lim sup‘𝐹) ≤ 𝐴))
35	1, 34	mpd 15	1 ⊢ (𝜑 → (lim sup‘𝐹) ≤ 𝐴)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065  Vcvv 3432   ∩ cin 3886   ⊆ wss 3887   class class class wbr 5074   ↦ cmpt 5157   “ cima 5592  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275  supcsup 9199  ℝcr 10870  +∞cpnf 11006  ℝ^*cxr 11008   < clt 11009   ≤ cle 11010  [,)cico 13081  lim supclsp 15179

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-po 5503  df-so 5504  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-sup 9201  df-inf 9202  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-ico 13085  df-limsup 15180

This theorem is referenced by:  caucvgrlem  15384  limsupre  43182  limsupbnd1f  43227