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Theorem limsupbnd1 15364
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 485 . . . . 5 ((𝜑𝑘 ∈ ℝ) → 𝑘 ∈ ℝ)
7 limsupbnd.3 . . . . . 6 (𝜑𝐴 ∈ ℝ*)
87adantr 481 . . . . 5 ((𝜑𝑘 ∈ ℝ) → 𝐴 ∈ ℝ*)
9 eqid 2736 . . . . . 6 (𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))
109limsupgle 15359 . . . . 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 1376 . . . 4 ((𝜑𝑘 ∈ ℝ) → (((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑘) ≤ 𝐴 ↔ ∀𝑗𝐵 (𝑘𝑗 → (𝐹𝑗) ≤ 𝐴)))
12 reex 11142 . . . . . . . . . . . 12 ℝ ∈ V
1312ssex 5278 . . . . . . . . . . 11 (𝐵 ⊆ ℝ → 𝐵 ∈ V)
142, 13syl 17 . . . . . . . . . 10 (𝜑𝐵 ∈ V)
15 xrex 12912 . . . . . . . . . . 11 * ∈ V
1615a1i 11 . . . . . . . . . 10 (𝜑 → ℝ* ∈ V)
17 fex2 7870 . . . . . . . . . 10 ((𝐹:𝐵⟶ℝ*𝐵 ∈ V ∧ ℝ* ∈ V) → 𝐹 ∈ V)
184, 14, 16, 17syl3anc 1371 . . . . . . . . 9 (𝜑𝐹 ∈ V)
19 limsupcl 15355 . . . . . . . . 9 (𝐹 ∈ V → (lim sup‘𝐹) ∈ ℝ*)
2018, 19syl 17 . . . . . . . 8 (𝜑 → (lim sup‘𝐹) ∈ ℝ*)
2120xrleidd 13071 . . . . . . 7 (𝜑 → (lim sup‘𝐹) ≤ (lim sup‘𝐹))
229limsuple 15360 . . . . . . . 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)))
232, 4, 20, 22syl3anc 1371 . . . . . . 7 (𝜑 → ((lim sup‘𝐹) ≤ (lim sup‘𝐹) ↔ ∀𝑘 ∈ ℝ (lim sup‘𝐹) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑘)))
2421, 23mpbid 231 . . . . . 6 (𝜑 → ∀𝑘 ∈ ℝ (lim sup‘𝐹) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑘))
2524r19.21bi 3234 . . . . 5 ((𝜑𝑘 ∈ ℝ) → (lim sup‘𝐹) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑘))
2620adantr 481 . . . . . 6 ((𝜑𝑘 ∈ ℝ) → (lim sup‘𝐹) ∈ ℝ*)
279limsupgf 15357 . . . . . . . 8 (𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < )):ℝ⟶ℝ*
2827a1i 11 . . . . . . 7 (𝜑 → (𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < )):ℝ⟶ℝ*)
2928ffvelcdmda 7035 . . . . . 6 ((𝜑𝑘 ∈ ℝ) → ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑘) ∈ ℝ*)
30 xrletr 13077 . . . . . 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))
3126, 29, 8, 30syl3anc 1371 . . . . 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))
3225, 31mpand 693 . . . 4 ((𝜑𝑘 ∈ ℝ) → (((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑘) ≤ 𝐴 → (lim sup‘𝐹) ≤ 𝐴))
3311, 32sylbird 259 . . 3 ((𝜑𝑘 ∈ ℝ) → (∀𝑗𝐵 (𝑘𝑗 → (𝐹𝑗) ≤ 𝐴) → (lim sup‘𝐹) ≤ 𝐴))
3433rexlimdva 3152 . 2 (𝜑 → (∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (𝐹𝑗) ≤ 𝐴) → (lim sup‘𝐹) ≤ 𝐴))
351, 34mpd 15 1 (𝜑 → (lim sup‘𝐹) ≤ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wcel 2106  wral 3064  wrex 3073  Vcvv 3445  cin 3909  wss 3910   class class class wbr 5105  cmpt 5188  cima 5636  wf 6492  cfv 6496  (class class class)co 7357  supcsup 9376  cr 11050  +∞cpnf 11186  *cxr 11188   < clt 11189  cle 11190  [,)cico 13266  lim supclsp 15352
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-po 5545  df-so 5546  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-sup 9378  df-inf 9379  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-ico 13270  df-limsup 15353
This theorem is referenced by:  caucvgrlem  15557  limsupre  43872  limsupbnd1f  43917
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