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Mirrors > Home > ILE Home > Th. List > climrecvg1n | Unicode version |
Description: A Cauchy sequence of real numbers converges, existence version. The rate of convergence is fixed: all terms after the nth term must be within of the nth term, where is a constant multiplier. (Contributed by Jim Kingdon, 23-Aug-2021.) |
Ref | Expression |
---|---|
climrecvg1n.f | |
climrecvg1n.c | |
climrecvg1n.cau |
Ref | Expression |
---|---|
climrecvg1n |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | climrecvg1n.f | . . 3 | |
2 | climrecvg1n.c | . . 3 | |
3 | climrecvg1n.cau | . . . . . . . . 9 | |
4 | 3 | r19.21bi 2449 | . . . . . . . 8 |
5 | 4 | r19.21bi 2449 | . . . . . . 7 |
6 | 1 | ad2antrr 471 | . . . . . . . . 9 |
7 | eluznn 8687 | . . . . . . . . . 10 | |
8 | 7 | adantll 459 | . . . . . . . . 9 |
9 | 6, 8 | ffvelrnd 5324 | . . . . . . . 8 |
10 | simplr 496 | . . . . . . . . 9 | |
11 | 6, 10 | ffvelrnd 5324 | . . . . . . . 8 |
12 | 2 | ad2antrr 471 | . . . . . . . . . 10 |
13 | 10 | nnrpd 8772 | . . . . . . . . . 10 |
14 | 12, 13 | rpdivcld 8791 | . . . . . . . . 9 |
15 | 14 | rpred 8773 | . . . . . . . 8 |
16 | 9, 11, 15 | absdifltd 10064 | . . . . . . 7 |
17 | 5, 16 | mpbid 145 | . . . . . 6 |
18 | 11, 15, 9 | ltsubaddd 7641 | . . . . . . 7 |
19 | 18 | anbi1d 452 | . . . . . 6 |
20 | 17, 19 | mpbid 145 | . . . . 5 |
21 | 20 | ralrimiva 2434 | . . . 4 |
22 | 21 | ralrimiva 2434 | . . 3 |
23 | 1, 2, 22 | cvg1n 9872 | . 2 |
24 | 1 | adantr 270 | . . . . . . . . . . . 12 |
25 | 24 | ad3antrrr 475 | . . . . . . . . . . 11 |
26 | eluznn 8687 | . . . . . . . . . . . 12 | |
27 | 26 | adantll 459 | . . . . . . . . . . 11 |
28 | 25, 27 | ffvelrnd 5324 | . . . . . . . . . 10 |
29 | simpr 108 | . . . . . . . . . . 11 | |
30 | 29 | ad3antrrr 475 | . . . . . . . . . 10 |
31 | simpllr 500 | . . . . . . . . . . 11 | |
32 | 31 | rpred 8773 | . . . . . . . . . 10 |
33 | 28, 30, 32 | absdifltd 10064 | . . . . . . . . 9 |
34 | 30, 32, 28 | ltsubaddd 7641 | . . . . . . . . . 10 |
35 | 34 | anbi1d 452 | . . . . . . . . 9 |
36 | 33, 35 | bitrd 186 | . . . . . . . 8 |
37 | ancom 262 | . . . . . . . 8 | |
38 | 36, 37 | syl6bb 194 | . . . . . . 7 |
39 | 38 | ralbidva 2364 | . . . . . 6 |
40 | 39 | rexbidva 2365 | . . . . 5 |
41 | 40 | ralbidva 2364 | . . . 4 |
42 | nnuz 8654 | . . . . . 6 | |
43 | 1zzd 8378 | . . . . . 6 | |
44 | nnex 8045 | . . . . . . . 8 | |
45 | 44 | a1i 9 | . . . . . . 7 |
46 | reex 7107 | . . . . . . . 8 | |
47 | 46 | a1i 9 | . . . . . . 7 |
48 | fex2 5079 | . . . . . . 7 | |
49 | 24, 45, 47, 48 | syl3anc 1169 | . . . . . 6 |
50 | eqidd 2082 | . . . . . 6 | |
51 | 29 | recnd 7147 | . . . . . 6 |
52 | 24 | ffvelrnda 5323 | . . . . . . 7 |
53 | 52 | recnd 7147 | . . . . . 6 |
54 | 42, 43, 49, 50, 51, 53 | clim2c 10123 | . . . . 5 |
55 | climrel 10119 | . . . . . 6 | |
56 | 55 | releldmi 4591 | . . . . 5 |
57 | 54, 56 | syl6bir 162 | . . . 4 |
58 | 41, 57 | sylbird 168 | . . 3 |
59 | 58 | impr 371 | . 2 |
60 | 23, 59 | rexlimddv 2481 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wcel 1433 wral 2348 wrex 2349 cvv 2601 class class class wbr 3785 cdm 4363 wf 4918 cfv 4922 (class class class)co 5532 cr 6980 c1 6982 caddc 6984 clt 7153 cmin 7279 cdiv 7760 cn 8039 cuz 8619 crp 8734 cabs 9883 cli 10117 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-mulrcl 7075 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-precex 7086 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-apti 7091 ax-pre-ltadd 7092 ax-pre-mulgt0 7093 ax-pre-mulext 7094 ax-arch 7095 ax-caucvg 7096 |
This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rmo 2356 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-if 3352 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-id 4048 df-po 4051 df-iso 4052 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-frec 6001 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-reap 7675 df-ap 7682 df-div 7761 df-inn 8040 df-2 8098 df-3 8099 df-4 8100 df-n0 8289 df-z 8352 df-uz 8620 df-rp 8735 df-iseq 9432 df-iexp 9476 df-cj 9729 df-re 9730 df-im 9731 df-rsqrt 9884 df-abs 9885 df-clim 10118 |
This theorem is referenced by: climcvg1nlem 10186 |
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