Metamath Proof Explorer


Theorem supcnvlimsup

Description: If a function on a set of upper integers has a real superior limit, the supremum of the rightmost parts of the function, converges to that superior limit. (Contributed by Glauco Siliprandi, 23-Oct-2021)

Ref Expression
Hypotheses supcnvlimsup.m φ M
supcnvlimsup.z Z = M
supcnvlimsup.f φ F : Z
supcnvlimsup.r φ lim sup F
Assertion supcnvlimsup φ k Z sup ran F k * < lim sup F

Proof

Step Hyp Ref Expression
1 supcnvlimsup.m φ M
2 supcnvlimsup.z Z = M
3 supcnvlimsup.f φ F : Z
4 supcnvlimsup.r φ lim sup F
5 3 adantr φ n Z F : Z
6 id n Z n Z
7 2 6 uzssd2 n Z n Z
8 7 adantl φ n Z n Z
9 5 8 feqresmpt φ n Z F n = m n F m
10 9 rneqd φ n Z ran F n = ran m n F m
11 10 supeq1d φ n Z sup ran F n * < = sup ran m n F m * <
12 nfcv _ m F
13 4 renepnfd φ lim sup F +∞
14 12 2 3 13 limsupubuz φ x m Z F m x
15 14 adantr φ n Z x m Z F m x
16 ssralv n Z m Z F m x m n F m x
17 7 16 syl n Z m Z F m x m n F m x
18 17 adantl φ n Z m Z F m x m n F m x
19 18 reximdv φ n Z x m Z F m x x m n F m x
20 15 19 mpd φ n Z x m n F m x
21 nfv m φ n Z
22 2 eluzelz2 n Z n
23 uzid n n n
24 ne0i n n n
25 22 23 24 3syl n Z n
26 25 adantl φ n Z n
27 5 adantr φ n Z m n F : Z
28 8 sselda φ n Z m n m Z
29 27 28 ffvelcdmd φ n Z m n F m
30 21 26 29 supxrre3rnmpt φ n Z sup ran m n F m * < x m n F m x
31 20 30 mpbird φ n Z sup ran m n F m * <
32 11 31 eqeltrd φ n Z sup ran F n * <
33 32 fmpttd φ n Z sup ran F n * < : Z
34 eqid i = i
35 2 eluzelz2 i Z i
36 35 peano2zd i Z i + 1
37 35 zred i Z i
38 lep1 i i i + 1
39 37 38 syl i Z i i + 1
40 34 35 36 39 eluzd i Z i + 1 i
41 uzss i + 1 i i + 1 i
42 ssres2 i + 1 i F i + 1 F i
43 rnss F i + 1 F i ran F i + 1 ran F i
44 40 41 42 43 4syl i Z ran F i + 1 ran F i
45 44 adantl φ i Z ran F i + 1 ran F i
46 rnresss ran F i ran F
47 46 a1i φ i Z ran F i ran F
48 3 frnd φ ran F
49 48 adantr φ i Z ran F
50 47 49 sstrd φ i Z ran F i
51 ressxr *
52 51 a1i φ i Z *
53 50 52 sstrd φ i Z ran F i *
54 supxrss ran F i + 1 ran F i ran F i * sup ran F i + 1 * < sup ran F i * <
55 45 53 54 syl2anc φ i Z sup ran F i + 1 * < sup ran F i * <
56 eqidd i Z n Z sup ran F n * < = n Z sup ran F n * <
57 fveq2 n = i + 1 n = i + 1
58 57 reseq2d n = i + 1 F n = F i + 1
59 58 rneqd n = i + 1 ran F n = ran F i + 1
60 59 supeq1d n = i + 1 sup ran F n * < = sup ran F i + 1 * <
61 60 adantl i Z n = i + 1 sup ran F n * < = sup ran F i + 1 * <
62 2 peano2uzs i Z i + 1 Z
63 xrltso < Or *
64 63 supex sup ran F i + 1 * < V
65 64 a1i i Z sup ran F i + 1 * < V
66 56 61 62 65 fvmptd i Z n Z sup ran F n * < i + 1 = sup ran F i + 1 * <
67 66 adantl φ i Z n Z sup ran F n * < i + 1 = sup ran F i + 1 * <
68 fveq2 n = i n = i
69 68 reseq2d n = i F n = F i
70 69 rneqd n = i ran F n = ran F i
71 70 supeq1d n = i sup ran F n * < = sup ran F i * <
72 71 adantl i Z n = i sup ran F n * < = sup ran F i * <
73 id i Z i Z
74 63 supex sup ran F i * < V
75 74 a1i i Z sup ran F i * < V
76 56 72 73 75 fvmptd i Z n Z sup ran F n * < i = sup ran F i * <
77 76 adantl φ i Z n Z sup ran F n * < i = sup ran F i * <
78 67 77 breq12d φ i Z n Z sup ran F n * < i + 1 n Z sup ran F n * < i sup ran F i + 1 * < sup ran F i * <
79 55 78 mpbird φ i Z n Z sup ran F n * < i + 1 n Z sup ran F n * < i
80 nfcv _ j F
81 3 frexr φ F : Z *
82 80 1 2 81 limsupre3uz φ lim sup F x i Z j i x F j x i Z j i F j x
83 4 82 mpbid φ x i Z j i x F j x i Z j i F j x
84 83 simpld φ x i Z j i x F j
85 simp-4r φ x i Z j i x F j x
86 85 rexrd φ x i Z j i x F j x *
87 81 3ad2ant1 φ i Z j i F : Z *
88 2 uztrn2 i Z j i j Z
89 88 3adant1 φ i Z j i j Z
90 87 89 ffvelcdmd φ i Z j i F j *
91 90 ad5ant134 φ x i Z j i x F j F j *
92 53 supxrcld φ i Z sup ran F i * < *
93 92 ad5ant13 φ x i Z j i x F j sup ran F i * < *
94 simpr φ x i Z j i x F j x F j
95 53 3adant3 φ i Z j i ran F i *
96 fvres j i F i j = F j
97 96 eqcomd j i F j = F i j
98 97 3ad2ant3 φ i Z j i F j = F i j
99 3 ffnd φ F Fn Z
100 99 adantr φ i Z F Fn Z
101 2 73 uzssd2 i Z i Z
102 101 adantl φ i Z i Z
103 fnssres F Fn Z i Z F i Fn i
104 100 102 103 syl2anc φ i Z F i Fn i
105 104 3adant3 φ i Z j i F i Fn i
106 simp3 φ i Z j i j i
107 fnfvelrn F i Fn i j i F i j ran F i
108 105 106 107 syl2anc φ i Z j i F i j ran F i
109 98 108 eqeltrd φ i Z j i F j ran F i
110 eqid sup ran F i * < = sup ran F i * <
111 95 109 110 supxrubd φ i Z j i F j sup ran F i * <
112 111 ad5ant134 φ x i Z j i x F j F j sup ran F i * <
113 86 91 93 94 112 xrletrd φ x i Z j i x F j x sup ran F i * <
114 113 rexlimdva2 φ x i Z j i x F j x sup ran F i * <
115 114 ralimdva φ x i Z j i x F j i Z x sup ran F i * <
116 115 reximdva φ x i Z j i x F j x i Z x sup ran F i * <
117 84 116 mpd φ x i Z x sup ran F i * <
118 simpl y = x i Z y = x
119 76 adantl y = x i Z n Z sup ran F n * < i = sup ran F i * <
120 118 119 breq12d y = x i Z y n Z sup ran F n * < i x sup ran F i * <
121 120 ralbidva y = x i Z y n Z sup ran F n * < i i Z x sup ran F i * <
122 121 cbvrexvw y i Z y n Z sup ran F n * < i x i Z x sup ran F i * <
123 117 122 sylibr φ y i Z y n Z sup ran F n * < i
124 2 1 33 79 123 climinf φ n Z sup ran F n * < inf ran n Z sup ran F n * < <
125 fveq2 n = k n = k
126 125 reseq2d n = k F n = F k
127 126 rneqd n = k ran F n = ran F k
128 127 supeq1d n = k sup ran F n * < = sup ran F k * <
129 128 cbvmptv n Z sup ran F n * < = k Z sup ran F k * <
130 129 a1i φ n Z sup ran F n * < = k Z sup ran F k * <
131 1 2 3 4 limsupvaluz2 φ lim sup F = inf ran n Z sup ran F n * < <
132 131 eqcomd φ inf ran n Z sup ran F n * < < = lim sup F
133 130 132 breq12d φ n Z sup ran F n * < inf ran n Z sup ran F n * < < k Z sup ran F k * < lim sup F
134 124 133 mpbid φ k Z sup ran F k * < lim sup F