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Theorem limsupbnd1 14383
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 472 . . . . 5 ((𝜑𝑘 ∈ ℝ) → 𝐵 ⊆ ℝ)
4 limsupbnd.2 . . . . . 6 (𝜑𝐹:𝐵⟶ℝ*)
54adantr 472 . . . . 5 ((𝜑𝑘 ∈ ℝ) → 𝐹:𝐵⟶ℝ*)
6 simpr 479 . . . . 5 ((𝜑𝑘 ∈ ℝ) → 𝑘 ∈ ℝ)
7 limsupbnd.3 . . . . . 6 (𝜑𝐴 ∈ ℝ*)
87adantr 472 . . . . 5 ((𝜑𝑘 ∈ ℝ) → 𝐴 ∈ ℝ*)
9 eqid 2748 . . . . . 6 (𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))
109limsupgle 14378 . . . . 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 1469 . . . 4 ((𝜑𝑘 ∈ ℝ) → (((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑘) ≤ 𝐴 ↔ ∀𝑗𝐵 (𝑘𝑗 → (𝐹𝑗) ≤ 𝐴)))
12 reex 10190 . . . . . . . . . . . 12 ℝ ∈ V
1312ssex 4942 . . . . . . . . . . 11 (𝐵 ⊆ ℝ → 𝐵 ∈ V)
142, 13syl 17 . . . . . . . . . 10 (𝜑𝐵 ∈ V)
15 xrex 11993 . . . . . . . . . . 11 * ∈ V
1615a1i 11 . . . . . . . . . 10 (𝜑 → ℝ* ∈ V)
17 fex2 7274 . . . . . . . . . 10 ((𝐹:𝐵⟶ℝ*𝐵 ∈ V ∧ ℝ* ∈ V) → 𝐹 ∈ V)
184, 14, 16, 17syl3anc 1463 . . . . . . . . 9 (𝜑𝐹 ∈ V)
19 limsupcl 14374 . . . . . . . . 9 (𝐹 ∈ V → (lim sup‘𝐹) ∈ ℝ*)
2018, 19syl 17 . . . . . . . 8 (𝜑 → (lim sup‘𝐹) ∈ ℝ*)
21 xrleid 12147 . . . . . . . 8 ((lim sup‘𝐹) ∈ ℝ* → (lim sup‘𝐹) ≤ (lim sup‘𝐹))
2220, 21syl 17 . . . . . . 7 (𝜑 → (lim sup‘𝐹) ≤ (lim sup‘𝐹))
239limsuple 14379 . . . . . . . 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 1463 . . . . . . 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 3058 . . . . 5 ((𝜑𝑘 ∈ ℝ) → (lim sup‘𝐹) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑘))
2720adantr 472 . . . . . 6 ((𝜑𝑘 ∈ ℝ) → (lim sup‘𝐹) ∈ ℝ*)
289limsupgf 14376 . . . . . . . 8 (𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < )):ℝ⟶ℝ*
2928a1i 11 . . . . . . 7 (𝜑 → (𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < )):ℝ⟶ℝ*)
3029ffvelrnda 6510 . . . . . 6 ((𝜑𝑘 ∈ ℝ) → ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑘) ∈ ℝ*)
31 xrletr 12153 . . . . . 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 1463 . . . . 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 713 . . . 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 3157 . 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 383  wcel 2127  wral 3038  wrex 3039  Vcvv 3328  cin 3702  wss 3703   class class class wbr 4792  cmpt 4869  cima 5257  wf 6033  cfv 6037  (class class class)co 6801  supcsup 8499  cr 10098  +∞cpnf 10234  *cxr 10236   < clt 10237  cle 10238  [,)cico 12341  lim supclsp 14371
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102  ax-cnex 10155  ax-resscn 10156  ax-1cn 10157  ax-icn 10158  ax-addcl 10159  ax-addrcl 10160  ax-mulcl 10161  ax-mulrcl 10162  ax-mulcom 10163  ax-addass 10164  ax-mulass 10165  ax-distr 10166  ax-i2m1 10167  ax-1ne0 10168  ax-1rid 10169  ax-rnegex 10170  ax-rrecex 10171  ax-cnre 10172  ax-pre-lttri 10173  ax-pre-lttrn 10174  ax-pre-ltadd 10175  ax-pre-mulgt0 10176  ax-pre-sup 10177
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-nel 3024  df-ral 3043  df-rex 3044  df-reu 3045  df-rmo 3046  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-br 4793  df-opab 4853  df-mpt 4870  df-id 5162  df-po 5175  df-so 5176  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-f1 6042  df-fo 6043  df-f1o 6044  df-fv 6045  df-riota 6762  df-ov 6804  df-oprab 6805  df-mpt2 6806  df-er 7899  df-en 8110  df-dom 8111  df-sdom 8112  df-sup 8501  df-inf 8502  df-pnf 10239  df-mnf 10240  df-xr 10241  df-ltxr 10242  df-le 10243  df-sub 10431  df-neg 10432  df-ico 12345  df-limsup 14372
This theorem is referenced by:  caucvgrlem  14573  limsupre  40345  limsupbnd1f  40390
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