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Theorem limsupbnd1 15484
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 479 . . . . 5 ((𝜑𝑘 ∈ ℝ) → 𝐵 ⊆ ℝ)
4 limsupbnd.2 . . . . . 6 (𝜑𝐹:𝐵⟶ℝ*)
54adantr 479 . . . . 5 ((𝜑𝑘 ∈ ℝ) → 𝐹:𝐵⟶ℝ*)
6 simpr 483 . . . . 5 ((𝜑𝑘 ∈ ℝ) → 𝑘 ∈ ℝ)
7 limsupbnd.3 . . . . . 6 (𝜑𝐴 ∈ ℝ*)
87adantr 479 . . . . 5 ((𝜑𝑘 ∈ ℝ) → 𝐴 ∈ ℝ*)
9 eqid 2726 . . . . . 6 (𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))
109limsupgle 15479 . . . . 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 1373 . . . 4 ((𝜑𝑘 ∈ ℝ) → (((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑘) ≤ 𝐴 ↔ ∀𝑗𝐵 (𝑘𝑗 → (𝐹𝑗) ≤ 𝐴)))
12 reex 11249 . . . . . . . . . . . 12 ℝ ∈ V
1312ssex 5326 . . . . . . . . . . 11 (𝐵 ⊆ ℝ → 𝐵 ∈ V)
142, 13syl 17 . . . . . . . . . 10 (𝜑𝐵 ∈ V)
15 xrex 13023 . . . . . . . . . . 11 * ∈ V
1615a1i 11 . . . . . . . . . 10 (𝜑 → ℝ* ∈ V)
17 fex2 7947 . . . . . . . . . 10 ((𝐹:𝐵⟶ℝ*𝐵 ∈ V ∧ ℝ* ∈ V) → 𝐹 ∈ V)
184, 14, 16, 17syl3anc 1368 . . . . . . . . 9 (𝜑𝐹 ∈ V)
19 limsupcl 15475 . . . . . . . . 9 (𝐹 ∈ V → (lim sup‘𝐹) ∈ ℝ*)
2018, 19syl 17 . . . . . . . 8 (𝜑 → (lim sup‘𝐹) ∈ ℝ*)
2120xrleidd 13185 . . . . . . 7 (𝜑 → (lim sup‘𝐹) ≤ (lim sup‘𝐹))
229limsuple 15480 . . . . . . . 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 1368 . . . . . . 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 3239 . . . . 5 ((𝜑𝑘 ∈ ℝ) → (lim sup‘𝐹) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑘))
2620adantr 479 . . . . . 6 ((𝜑𝑘 ∈ ℝ) → (lim sup‘𝐹) ∈ ℝ*)
279limsupgf 15477 . . . . . . . 8 (𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < )):ℝ⟶ℝ*
2827a1i 11 . . . . . . 7 (𝜑 → (𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < )):ℝ⟶ℝ*)
2928ffvelcdmda 7098 . . . . . 6 ((𝜑𝑘 ∈ ℝ) → ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑘) ∈ ℝ*)
30 xrletr 13191 . . . . . 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 1368 . . . . 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 3145 . 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 394  wcel 2099  wral 3051  wrex 3060  Vcvv 3462  cin 3946  wss 3947   class class class wbr 5153  cmpt 5236  cima 5685  wf 6550  cfv 6554  (class class class)co 7424  supcsup 9483  cr 11157  +∞cpnf 11295  *cxr 11297   < clt 11298  cle 11299  [,)cico 13380  lim supclsp 15472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11214  ax-resscn 11215  ax-1cn 11216  ax-icn 11217  ax-addcl 11218  ax-addrcl 11219  ax-mulcl 11220  ax-mulrcl 11221  ax-mulcom 11222  ax-addass 11223  ax-mulass 11224  ax-distr 11225  ax-i2m1 11226  ax-1ne0 11227  ax-1rid 11228  ax-rnegex 11229  ax-rrecex 11230  ax-cnre 11231  ax-pre-lttri 11232  ax-pre-lttrn 11233  ax-pre-ltadd 11234  ax-pre-mulgt0 11235  ax-pre-sup 11236
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-po 5594  df-so 5595  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-er 8734  df-en 8975  df-dom 8976  df-sdom 8977  df-sup 9485  df-inf 9486  df-pnf 11300  df-mnf 11301  df-xr 11302  df-ltxr 11303  df-le 11304  df-sub 11496  df-neg 11497  df-ico 13384  df-limsup 15473
This theorem is referenced by:  caucvgrlem  15677  limsupre  45262  limsupbnd1f  45307
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