Theorem limsupbnd1 14061

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 476	. . . . 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 2610	. . . . . 6 ⊢ (𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))
10	9	limsupgle 14056	. . . . 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 1324	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℝ) → (((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑘) ≤ 𝐴 ↔ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴)))
12		reex 9906	. . . . . . . . . . . 12 ⊢ ℝ ∈ V
13	12	ssex 4730	. . . . . . . . . . 11 ⊢ (𝐵 ⊆ ℝ → 𝐵 ∈ V)
14	2, 13	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ V)
15		xrex 11705	. . . . . . . . . . 11 ⊢ ℝ* ∈ V
16	15	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ℝ* ∈ V)
17		fex2 7014	. . . . . . . . . 10 ⊢ ((𝐹:𝐵⟶ℝ* ∧ 𝐵 ∈ V ∧ ℝ* ∈ V) → 𝐹 ∈ V)
18	4, 14, 16, 17	syl3anc 1318	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ V)
19		limsupcl 14052	. . . . . . . . 9 ⊢ (𝐹 ∈ V → (lim sup‘𝐹) ∈ ℝ*)
20	18, 19	syl 17	. . . . . . . 8 ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ*)
21		xrleid 11859	. . . . . . . 8 ⊢ ((lim sup‘𝐹) ∈ ℝ* → (lim sup‘𝐹) ≤ (lim sup‘𝐹))
22	20, 21	syl 17	. . . . . . 7 ⊢ (𝜑 → (lim sup‘𝐹) ≤ (lim sup‘𝐹))
23	9	limsuple 14057	. . . . . . . 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)))
24	2, 4, 20, 23	syl3anc 1318	. . . . . . 7 ⊢ (𝜑 → ((lim sup‘𝐹) ≤ (lim sup‘𝐹) ↔ ∀𝑘 ∈ ℝ (lim sup‘𝐹) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑘)))
25	22, 24	mpbid 221	. . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ ℝ (lim sup‘𝐹) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑘))
26	25	r19.21bi 2916	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℝ) → (lim sup‘𝐹) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑘))
27	20	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℝ) → (lim sup‘𝐹) ∈ ℝ*)
28	9	limsupgf 14054	. . . . . . . 8 ⊢ (𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < )):ℝ⟶ℝ*
29	28	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < )):ℝ⟶ℝ*)
30	29	ffvelrnda 6267	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℝ) → ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑘) ∈ ℝ*)
31		xrletr 11865	. . . . . 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))
32	27, 30, 8, 31	syl3anc 1318	. . . . 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))
33	26, 32	mpand 707	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℝ) → (((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑘) ≤ 𝐴 → (lim sup‘𝐹) ≤ 𝐴))
34	11, 33	sylbird 249	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℝ) → (∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴) → (lim sup‘𝐹) ≤ 𝐴))
35	34	rexlimdva 3013	. 2 ⊢ (𝜑 → (∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴) → (lim sup‘𝐹) ≤ 𝐴))
36	1, 35	mpd 15	1 ⊢ (𝜑 → (lim sup‘𝐹) ≤ 𝐴)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540   class class class wbr 4583   ↦ cmpt 4643   “ cima 5041  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  supcsup 8229  ℝcr 9814  +∞cpnf 9950  ℝ^*cxr 9952   < clt 9953   ≤ cle 9954  [,)cico 12048  lim supclsp 14049

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-po 4959  df-so 4960  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-sup 8231  df-inf 8232  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-ico 12052  df-limsup 14050

This theorem is referenced by:  caucvgrlem  14251  limsupre  38708