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Theorem limsupbnd1 14147
 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 (𝜑𝐵 ⊆ ℝ)
limsupbnd.2 (𝜑𝐹:𝐵⟶ℝ*)
limsupbnd.3 (𝜑𝐴 ∈ ℝ*)
limsupbnd1.4 (𝜑 → ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (𝐹𝑗) ≤ 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.
StepHypRef Expression
1 limsupbnd1.4 . 2 (𝜑 → ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (𝐹𝑗) ≤ 𝐴))
2 limsupbnd.1 . . . . . 6 (𝜑𝐵 ⊆ ℝ)
32adantr 481 . . . . 5 ((𝜑𝑘 ∈ ℝ) → 𝐵 ⊆ ℝ)
4 limsupbnd.2 . . . . . 6 (𝜑𝐹:𝐵⟶ℝ*)
54adantr 481 . . . . 5 ((𝜑𝑘 ∈ ℝ) → 𝐹:𝐵⟶ℝ*)
6 simpr 477 . . . . 5 ((𝜑𝑘 ∈ ℝ) → 𝑘 ∈ ℝ)
7 limsupbnd.3 . . . . . 6 (𝜑𝐴 ∈ ℝ*)
87adantr 481 . . . . 5 ((𝜑𝑘 ∈ ℝ) → 𝐴 ∈ ℝ*)
9 eqid 2621 . . . . . 6 (𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))
109limsupgle 14142 . . . . 5 (((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ*) ∧ 𝑘 ∈ ℝ ∧ 𝐴 ∈ ℝ*) → (((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑘) ≤ 𝐴 ↔ ∀𝑗𝐵 (𝑘𝑗 → (𝐹𝑗) ≤ 𝐴)))
113, 5, 6, 8, 10syl211anc 1329 . . . 4 ((𝜑𝑘 ∈ ℝ) → (((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑘) ≤ 𝐴 ↔ ∀𝑗𝐵 (𝑘𝑗 → (𝐹𝑗) ≤ 𝐴)))
12 reex 9971 . . . . . . . . . . . 12 ℝ ∈ V
1312ssex 4762 . . . . . . . . . . 11 (𝐵 ⊆ ℝ → 𝐵 ∈ V)
142, 13syl 17 . . . . . . . . . 10 (𝜑𝐵 ∈ V)
15 xrex 11773 . . . . . . . . . . 11 * ∈ V
1615a1i 11 . . . . . . . . . 10 (𝜑 → ℝ* ∈ V)
17 fex2 7068 . . . . . . . . . 10 ((𝐹:𝐵⟶ℝ*𝐵 ∈ V ∧ ℝ* ∈ V) → 𝐹 ∈ V)
184, 14, 16, 17syl3anc 1323 . . . . . . . . 9 (𝜑𝐹 ∈ V)
19 limsupcl 14138 . . . . . . . . 9 (𝐹 ∈ V → (lim sup‘𝐹) ∈ ℝ*)
2018, 19syl 17 . . . . . . . 8 (𝜑 → (lim sup‘𝐹) ∈ ℝ*)
21 xrleid 11927 . . . . . . . 8 ((lim sup‘𝐹) ∈ ℝ* → (lim sup‘𝐹) ≤ (lim sup‘𝐹))
2220, 21syl 17 . . . . . . 7 (𝜑 → (lim sup‘𝐹) ≤ (lim sup‘𝐹))
239limsuple 14143 . . . . . . . 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)))
242, 4, 20, 23syl3anc 1323 . . . . . . 7 (𝜑 → ((lim sup‘𝐹) ≤ (lim sup‘𝐹) ↔ ∀𝑘 ∈ ℝ (lim sup‘𝐹) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑘)))
2522, 24mpbid 222 . . . . . 6 (𝜑 → ∀𝑘 ∈ ℝ (lim sup‘𝐹) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑘))
2625r19.21bi 2927 . . . . 5 ((𝜑𝑘 ∈ ℝ) → (lim sup‘𝐹) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑘))
2720adantr 481 . . . . . 6 ((𝜑𝑘 ∈ ℝ) → (lim sup‘𝐹) ∈ ℝ*)
289limsupgf 14140 . . . . . . . 8 (𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < )):ℝ⟶ℝ*
2928a1i 11 . . . . . . 7 (𝜑 → (𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < )):ℝ⟶ℝ*)
3029ffvelrnda 6315 . . . . . 6 ((𝜑𝑘 ∈ ℝ) → ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑘) ∈ ℝ*)
31 xrletr 11933 . . . . . 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))
3227, 30, 8, 31syl3anc 1323 . . . . 5 ((𝜑𝑘 ∈ 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))
3326, 32mpand 710 . . . 4 ((𝜑𝑘 ∈ ℝ) → (((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑘) ≤ 𝐴 → (lim sup‘𝐹) ≤ 𝐴))
3411, 33sylbird 250 . . 3 ((𝜑𝑘 ∈ ℝ) → (∀𝑗𝐵 (𝑘𝑗 → (𝐹𝑗) ≤ 𝐴) → (lim sup‘𝐹) ≤ 𝐴))
3534rexlimdva 3024 . 2 (𝜑 → (∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (𝐹𝑗) ≤ 𝐴) → (lim sup‘𝐹) ≤ 𝐴))
361, 35mpd 15 1 (𝜑 → (lim sup‘𝐹) ≤ 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∈ wcel 1987  ∀wral 2907  ∃wrex 2908  Vcvv 3186   ∩ cin 3554   ⊆ wss 3555   class class class wbr 4613   ↦ cmpt 4673   “ cima 5077  ⟶wf 5843  ‘cfv 5847  (class class class)co 6604  supcsup 8290  ℝcr 9879  +∞cpnf 10015  ℝ*cxr 10017   < clt 10018   ≤ cle 10019  [,)cico 12119  lim supclsp 14135 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-pre-sup 9958 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-po 4995  df-so 4996  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-sup 8292  df-inf 8293  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-ico 12123  df-limsup 14136 This theorem is referenced by:  caucvgrlem  14337  limsupre  39274  limsupbnd1f  39319
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