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Mirrors > Home > MPE Home > Th. List > climserle | Structured version Visualization version GIF version |
Description: The partial sums of a converging infinite series with nonnegative terms are bounded by its limit. (Contributed by NM, 27-Dec-2005.) (Revised by Mario Carneiro, 9-Feb-2014.) |
Ref | Expression |
---|---|
clim2ser.1 | β’ π = (β€β₯βπ) |
climserle.2 | β’ (π β π β π) |
climserle.3 | β’ (π β seqπ( + , πΉ) β π΄) |
climserle.4 | β’ ((π β§ π β π) β (πΉβπ) β β) |
climserle.5 | β’ ((π β§ π β π) β 0 β€ (πΉβπ)) |
Ref | Expression |
---|---|
climserle | β’ (π β (seqπ( + , πΉ)βπ) β€ π΄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clim2ser.1 | . 2 β’ π = (β€β₯βπ) | |
2 | climserle.2 | . 2 β’ (π β π β π) | |
3 | climserle.3 | . 2 β’ (π β seqπ( + , πΉ) β π΄) | |
4 | 2, 1 | eleqtrdi 2837 | . . . . 5 β’ (π β π β (β€β₯βπ)) |
5 | eluzel2 12831 | . . . . 5 β’ (π β (β€β₯βπ) β π β β€) | |
6 | 4, 5 | syl 17 | . . . 4 β’ (π β π β β€) |
7 | climserle.4 | . . . 4 β’ ((π β§ π β π) β (πΉβπ) β β) | |
8 | 1, 6, 7 | serfre 14002 | . . 3 β’ (π β seqπ( + , πΉ):πβΆβ) |
9 | 8 | ffvelcdmda 7080 | . 2 β’ ((π β§ π β π) β (seqπ( + , πΉ)βπ) β β) |
10 | 1 | peano2uzs 12890 | . . . . 5 β’ (π β π β (π + 1) β π) |
11 | fveq2 6885 | . . . . . . . . 9 β’ (π = (π + 1) β (πΉβπ) = (πΉβ(π + 1))) | |
12 | 11 | breq2d 5153 | . . . . . . . 8 β’ (π = (π + 1) β (0 β€ (πΉβπ) β 0 β€ (πΉβ(π + 1)))) |
13 | 12 | imbi2d 340 | . . . . . . 7 β’ (π = (π + 1) β ((π β 0 β€ (πΉβπ)) β (π β 0 β€ (πΉβ(π + 1))))) |
14 | climserle.5 | . . . . . . . 8 β’ ((π β§ π β π) β 0 β€ (πΉβπ)) | |
15 | 14 | expcom 413 | . . . . . . 7 β’ (π β π β (π β 0 β€ (πΉβπ))) |
16 | 13, 15 | vtoclga 3560 | . . . . . 6 β’ ((π + 1) β π β (π β 0 β€ (πΉβ(π + 1)))) |
17 | 16 | impcom 407 | . . . . 5 β’ ((π β§ (π + 1) β π) β 0 β€ (πΉβ(π + 1))) |
18 | 10, 17 | sylan2 592 | . . . 4 β’ ((π β§ π β π) β 0 β€ (πΉβ(π + 1))) |
19 | 11 | eleq1d 2812 | . . . . . . . . 9 β’ (π = (π + 1) β ((πΉβπ) β β β (πΉβ(π + 1)) β β)) |
20 | 19 | imbi2d 340 | . . . . . . . 8 β’ (π = (π + 1) β ((π β (πΉβπ) β β) β (π β (πΉβ(π + 1)) β β))) |
21 | 7 | expcom 413 | . . . . . . . 8 β’ (π β π β (π β (πΉβπ) β β)) |
22 | 20, 21 | vtoclga 3560 | . . . . . . 7 β’ ((π + 1) β π β (π β (πΉβ(π + 1)) β β)) |
23 | 22 | impcom 407 | . . . . . 6 β’ ((π β§ (π + 1) β π) β (πΉβ(π + 1)) β β) |
24 | 10, 23 | sylan2 592 | . . . . 5 β’ ((π β§ π β π) β (πΉβ(π + 1)) β β) |
25 | 9, 24 | addge01d 11806 | . . . 4 β’ ((π β§ π β π) β (0 β€ (πΉβ(π + 1)) β (seqπ( + , πΉ)βπ) β€ ((seqπ( + , πΉ)βπ) + (πΉβ(π + 1))))) |
26 | 18, 25 | mpbid 231 | . . 3 β’ ((π β§ π β π) β (seqπ( + , πΉ)βπ) β€ ((seqπ( + , πΉ)βπ) + (πΉβ(π + 1)))) |
27 | simpr 484 | . . . . 5 β’ ((π β§ π β π) β π β π) | |
28 | 27, 1 | eleqtrdi 2837 | . . . 4 β’ ((π β§ π β π) β π β (β€β₯βπ)) |
29 | seqp1 13987 | . . . 4 β’ (π β (β€β₯βπ) β (seqπ( + , πΉ)β(π + 1)) = ((seqπ( + , πΉ)βπ) + (πΉβ(π + 1)))) | |
30 | 28, 29 | syl 17 | . . 3 β’ ((π β§ π β π) β (seqπ( + , πΉ)β(π + 1)) = ((seqπ( + , πΉ)βπ) + (πΉβ(π + 1)))) |
31 | 26, 30 | breqtrrd 5169 | . 2 β’ ((π β§ π β π) β (seqπ( + , πΉ)βπ) β€ (seqπ( + , πΉ)β(π + 1))) |
32 | 1, 2, 3, 9, 31 | climub 15614 | 1 β’ (π β (seqπ( + , πΉ)βπ) β€ π΄) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 class class class wbr 5141 βcfv 6537 (class class class)co 7405 βcr 11111 0cc0 11112 1c1 11113 + caddc 11115 β€ cle 11253 β€cz 12562 β€β₯cuz 12826 seqcseq 13972 β cli 15434 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-er 8705 df-pm 8825 df-en 8942 df-dom 8943 df-sdom 8944 df-sup 9439 df-inf 9440 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-z 12563 df-uz 12827 df-rp 12981 df-fz 13491 df-fl 13763 df-seq 13973 df-exp 14033 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-clim 15438 df-rlim 15439 |
This theorem is referenced by: isumrpcl 15795 ege2le3 16040 prmreclem6 16863 ioombl1lem4 25445 rge0scvg 33459 |
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