Theorem limsupbnd1 15515

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	⊢ (𝜑 → 𝐵 ⊆ ℝ)
limsupbnd.2	⊢ (𝜑 → 𝐹:𝐵⟶ℝ*)
limsupbnd.3	⊢ (𝜑 → 𝐴 ∈ ℝ*)
limsupbnd1.4	⊢ (𝜑 → ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 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.

Step	Hyp	Ref	Expression
1		limsupbnd1.4	. 2 ⊢ (𝜑 → ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴))
2		limsupbnd.1	. . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ ℝ)
3	2	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℝ) → 𝐵 ⊆ ℝ)
4		limsupbnd.2	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐵⟶ℝ*)
5	4	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℝ) → 𝐹:𝐵⟶ℝ*)
6		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℝ) → 𝑘 ∈ ℝ)
7		limsupbnd.3	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ*)
8	7	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℝ) → 𝐴 ∈ ℝ*)
9		eqid 2735	. . . . . 6 ⊢ (𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))
10	9	limsupgle 15510	. . . . 5 ⊢ (((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ*) ∧ 𝑘 ∈ ℝ ∧ 𝐴 ∈ ℝ*) → (((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑘) ≤ 𝐴 ↔ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴)))
11	3, 5, 6, 8, 10	syl211anc 1375	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℝ) → (((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑘) ≤ 𝐴 ↔ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴)))
12		reex 11244	. . . . . . . . . . . 12 ⊢ ℝ ∈ V
13	12	ssex 5327	. . . . . . . . . . 11 ⊢ (𝐵 ⊆ ℝ → 𝐵 ∈ V)
14	2, 13	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ V)
15		xrex 13027	. . . . . . . . . . 11 ⊢ ℝ* ∈ V
16	15	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ℝ* ∈ V)
17		fex2 7957	. . . . . . . . . 10 ⊢ ((𝐹:𝐵⟶ℝ* ∧ 𝐵 ∈ V ∧ ℝ* ∈ V) → 𝐹 ∈ V)
18	4, 14, 16, 17	syl3anc 1370	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ V)
19		limsupcl 15506	. . . . . . . . 9 ⊢ (𝐹 ∈ V → (lim sup‘𝐹) ∈ ℝ*)
20	18, 19	syl 17	. . . . . . . 8 ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ*)
21	20	xrleidd 13191	. . . . . . 7 ⊢ (𝜑 → (lim sup‘𝐹) ≤ (lim sup‘𝐹))
22	9	limsuple 15511	. . . . . . . 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)))
23	2, 4, 20, 22	syl3anc 1370	. . . . . . 7 ⊢ (𝜑 → ((lim sup‘𝐹) ≤ (lim sup‘𝐹) ↔ ∀𝑘 ∈ ℝ (lim sup‘𝐹) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑘)))
24	21, 23	mpbid 232	. . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ ℝ (lim sup‘𝐹) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑘))
25	24	r19.21bi 3249	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℝ) → (lim sup‘𝐹) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑘))
26	20	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℝ) → (lim sup‘𝐹) ∈ ℝ*)
27	9	limsupgf 15508	. . . . . . . 8 ⊢ (𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < )):ℝ⟶ℝ*
28	27	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < )):ℝ⟶ℝ*)
29	28	ffvelcdmda 7104	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℝ) → ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑘) ∈ ℝ*)
30		xrletr 13197	. . . . . 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))
31	26, 29, 8, 30	syl3anc 1370	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 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))
32	25, 31	mpand 695	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℝ) → (((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑘) ≤ 𝐴 → (lim sup‘𝐹) ≤ 𝐴))
33	11, 32	sylbird 260	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℝ) → (∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴) → (lim sup‘𝐹) ≤ 𝐴))
34	33	rexlimdva 3153	. 2 ⊢ (𝜑 → (∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴) → (lim sup‘𝐹) ≤ 𝐴))
35	1, 34	mpd 15	1 ⊢ (𝜑 → (lim sup‘𝐹) ≤ 𝐴)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2106  ∀wral 3059  ∃wrex 3068  Vcvv 3478   ∩ cin 3962   ⊆ wss 3963   class class class wbr 5148   ↦ cmpt 5231   “ cima 5692  ⟶wf 6559  ‘cfv 6563  (class class class)co 7431  supcsup 9478  ℝcr 11152  +∞cpnf 11290  ℝ^*cxr 11292   < clt 11293   ≤ cle 11294  [,)cico 13386  lim supclsp 15503

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-po 5597  df-so 5598  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-sup 9480  df-inf 9481  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-ico 13390  df-limsup 15504

This theorem is referenced by:  caucvgrlem  15706  limsupre  45597  limsupbnd1f  45642