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Theorem limsupbnd1 14421
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 466 . . . . 5 ((𝜑𝑘 ∈ ℝ) → 𝐵 ⊆ ℝ)
4 limsupbnd.2 . . . . . 6 (𝜑𝐹:𝐵⟶ℝ*)
54adantr 466 . . . . 5 ((𝜑𝑘 ∈ ℝ) → 𝐹:𝐵⟶ℝ*)
6 simpr 471 . . . . 5 ((𝜑𝑘 ∈ ℝ) → 𝑘 ∈ ℝ)
7 limsupbnd.3 . . . . . 6 (𝜑𝐴 ∈ ℝ*)
87adantr 466 . . . . 5 ((𝜑𝑘 ∈ ℝ) → 𝐴 ∈ ℝ*)
9 eqid 2771 . . . . . 6 (𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))
109limsupgle 14416 . . . . 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 1482 . . . 4 ((𝜑𝑘 ∈ ℝ) → (((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑘) ≤ 𝐴 ↔ ∀𝑗𝐵 (𝑘𝑗 → (𝐹𝑗) ≤ 𝐴)))
12 reex 10229 . . . . . . . . . . . 12 ℝ ∈ V
1312ssex 4936 . . . . . . . . . . 11 (𝐵 ⊆ ℝ → 𝐵 ∈ V)
142, 13syl 17 . . . . . . . . . 10 (𝜑𝐵 ∈ V)
15 xrex 12032 . . . . . . . . . . 11 * ∈ V
1615a1i 11 . . . . . . . . . 10 (𝜑 → ℝ* ∈ V)
17 fex2 7268 . . . . . . . . . 10 ((𝐹:𝐵⟶ℝ*𝐵 ∈ V ∧ ℝ* ∈ V) → 𝐹 ∈ V)
184, 14, 16, 17syl3anc 1476 . . . . . . . . 9 (𝜑𝐹 ∈ V)
19 limsupcl 14412 . . . . . . . . 9 (𝐹 ∈ V → (lim sup‘𝐹) ∈ ℝ*)
2018, 19syl 17 . . . . . . . 8 (𝜑 → (lim sup‘𝐹) ∈ ℝ*)
21 xrleid 12188 . . . . . . . 8 ((lim sup‘𝐹) ∈ ℝ* → (lim sup‘𝐹) ≤ (lim sup‘𝐹))
2220, 21syl 17 . . . . . . 7 (𝜑 → (lim sup‘𝐹) ≤ (lim sup‘𝐹))
239limsuple 14417 . . . . . . . 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 1476 . . . . . . 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 3081 . . . . 5 ((𝜑𝑘 ∈ ℝ) → (lim sup‘𝐹) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑘))
2720adantr 466 . . . . . 6 ((𝜑𝑘 ∈ ℝ) → (lim sup‘𝐹) ∈ ℝ*)
289limsupgf 14414 . . . . . . . 8 (𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < )):ℝ⟶ℝ*
2928a1i 11 . . . . . . 7 (𝜑 → (𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < )):ℝ⟶ℝ*)
3029ffvelrnda 6502 . . . . . 6 ((𝜑𝑘 ∈ ℝ) → ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑘) ∈ ℝ*)
31 xrletr 12194 . . . . . 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 1476 . . . . 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 675 . . . 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 3179 . 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 382  wcel 2145  wral 3061  wrex 3062  Vcvv 3351  cin 3722  wss 3723   class class class wbr 4786  cmpt 4863  cima 5252  wf 6027  cfv 6031  (class class class)co 6793  supcsup 8502  cr 10137  +∞cpnf 10273  *cxr 10275   < clt 10276  cle 10277  [,)cico 12382  lim supclsp 14409
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215  ax-pre-sup 10216
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-po 5170  df-so 5171  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-er 7896  df-en 8110  df-dom 8111  df-sdom 8112  df-sup 8504  df-inf 8505  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-ico 12386  df-limsup 14410
This theorem is referenced by:  caucvgrlem  14611  limsupre  40391  limsupbnd1f  40436
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