Theorem limsupbnd1 15448

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 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℝ) → 𝐵 ⊆ ℝ)
4		limsupbnd.2	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐵⟶ℝ*)
5	4	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℝ) → 𝐹:𝐵⟶ℝ*)
6		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℝ) → 𝑘 ∈ ℝ)
7		limsupbnd.3	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ*)
8	7	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℝ) → 𝐴 ∈ ℝ*)
9		eqid 2729	. . . . . 6 ⊢ (𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))
10	9	limsupgle 15443	. . . . 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 1378	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℝ) → (((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑘) ≤ 𝐴 ↔ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴)))
12		reex 11159	. . . . . . . . . . . 12 ⊢ ℝ ∈ V
13	12	ssex 5276	. . . . . . . . . . 11 ⊢ (𝐵 ⊆ ℝ → 𝐵 ∈ V)
14	2, 13	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ V)
15		xrex 12946	. . . . . . . . . . 11 ⊢ ℝ* ∈ V
16	15	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ℝ* ∈ V)
17		fex2 7912	. . . . . . . . . 10 ⊢ ((𝐹:𝐵⟶ℝ* ∧ 𝐵 ∈ V ∧ ℝ* ∈ V) → 𝐹 ∈ V)
18	4, 14, 16, 17	syl3anc 1373	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ V)
19		limsupcl 15439	. . . . . . . . 9 ⊢ (𝐹 ∈ V → (lim sup‘𝐹) ∈ ℝ*)
20	18, 19	syl 17	. . . . . . . 8 ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ*)
21	20	xrleidd 13112	. . . . . . 7 ⊢ (𝜑 → (lim sup‘𝐹) ≤ (lim sup‘𝐹))
22	9	limsuple 15444	. . . . . . . 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 1373	. . . . . . 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 232	. . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ ℝ (lim sup‘𝐹) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑘))
25	24	r19.21bi 3229	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℝ) → (lim sup‘𝐹) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑘))
26	20	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℝ) → (lim sup‘𝐹) ∈ ℝ*)
27	9	limsupgf 15441	. . . . . . . 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	ffvelcdmda 7056	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℝ) → ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑘) ∈ ℝ*)
30		xrletr 13118	. . . . . 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 1373	. . . . 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 695	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℝ) → (((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑘) ≤ 𝐴 → (lim sup‘𝐹) ≤ 𝐴))
33	11, 32	sylbird 260	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℝ) → (∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴) → (lim sup‘𝐹) ≤ 𝐴))
34	33	rexlimdva 3134	. 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 206   ∧ wa 395   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  Vcvv 3447   ∩ cin 3913   ⊆ wss 3914   class class class wbr 5107   ↦ cmpt 5188   “ cima 5641  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387  supcsup 9391  ℝcr 11067  +∞cpnf 11205  ℝ^*cxr 11207   < clt 11208   ≤ cle 11209  [,)cico 13308  lim supclsp 15436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145  ax-pre-sup 11146

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-po 5546  df-so 5547  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-er 8671  df-en 8919  df-dom 8920  df-sdom 8921  df-sup 9393  df-inf 9394  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-ico 13312  df-limsup 15437

This theorem is referenced by:  caucvgrlem  15639  limsupre  45639  limsupbnd1f  45684