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Theorem harmonic 12630
Description: The harmonic series  H diverges. This fact follows from the stronger emcl 20833, which establishes that the harmonic series grows as  log n  +  gamma  + o(1), but this uses a more elementary method, attributed to Nicole Oresme (1323-1382). (Contributed by Mario Carneiro, 11-Jul-2014.)
Hypotheses
Ref Expression
harmonic.1  |-  F  =  ( n  e.  NN  |->  ( 1  /  n
) )
harmonic.2  |-  H  =  seq  1 (  +  ,  F )
Assertion
Ref Expression
harmonic  |-  -.  H  e.  dom  ~~>

Proof of Theorem harmonic
Dummy variables  k 
j are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nn0uz 10512 . . . 4  |-  NN0  =  ( ZZ>= `  0 )
2 0z 10285 . . . . 5  |-  0  e.  ZZ
32a1i 11 . . . 4  |-  ( H  e.  dom  ~~>  ->  0  e.  ZZ )
4 1ex 9078 . . . . . 6  |-  1  e.  _V
54fvconst2 5939 . . . . 5  |-  ( k  e.  NN0  ->  ( ( NN0  X.  { 1 } ) `  k
)  =  1 )
65adantl 453 . . . 4  |-  ( ( H  e.  dom  ~~>  /\  k  e.  NN0 )  ->  (
( NN0  X.  { 1 } ) `  k
)  =  1 )
7 1re 9082 . . . . 5  |-  1  e.  RR
87a1i 11 . . . 4  |-  ( ( H  e.  dom  ~~>  /\  k  e.  NN0 )  ->  1  e.  RR )
9 harmonic.2 . . . . . . 7  |-  H  =  seq  1 (  +  ,  F )
109eleq1i 2498 . . . . . 6  |-  ( H  e.  dom  ~~>  <->  seq  1
(  +  ,  F
)  e.  dom  ~~>  )
1110biimpi 187 . . . . 5  |-  ( H  e.  dom  ~~>  ->  seq  1 (  +  ,  F )  e.  dom  ~~>  )
12 oveq2 6081 . . . . . . . . 9  |-  ( n  =  k  ->  (
1  /  n )  =  ( 1  / 
k ) )
13 harmonic.1 . . . . . . . . 9  |-  F  =  ( n  e.  NN  |->  ( 1  /  n
) )
14 ovex 6098 . . . . . . . . 9  |-  ( 1  /  k )  e. 
_V
1512, 13, 14fvmpt 5798 . . . . . . . 8  |-  ( k  e.  NN  ->  ( F `  k )  =  ( 1  / 
k ) )
16 nnrecre 10028 . . . . . . . 8  |-  ( k  e.  NN  ->  (
1  /  k )  e.  RR )
1715, 16eqeltrd 2509 . . . . . . 7  |-  ( k  e.  NN  ->  ( F `  k )  e.  RR )
1817adantl 453 . . . . . 6  |-  ( ( H  e.  dom  ~~>  /\  k  e.  NN )  ->  ( F `  k )  e.  RR )
19 nnrp 10613 . . . . . . . . . 10  |-  ( k  e.  NN  ->  k  e.  RR+ )
2019rpreccld 10650 . . . . . . . . 9  |-  ( k  e.  NN  ->  (
1  /  k )  e.  RR+ )
2120rpge0d 10644 . . . . . . . 8  |-  ( k  e.  NN  ->  0  <_  ( 1  /  k
) )
2221, 15breqtrrd 4230 . . . . . . 7  |-  ( k  e.  NN  ->  0  <_  ( F `  k
) )
2322adantl 453 . . . . . 6  |-  ( ( H  e.  dom  ~~>  /\  k  e.  NN )  ->  0  <_  ( F `  k
) )
24 nnre 9999 . . . . . . . . . 10  |-  ( k  e.  NN  ->  k  e.  RR )
2524lep1d 9934 . . . . . . . . 9  |-  ( k  e.  NN  ->  k  <_  ( k  +  1 ) )
26 nngt0 10021 . . . . . . . . . 10  |-  ( k  e.  NN  ->  0  <  k )
27 peano2re 9231 . . . . . . . . . . 11  |-  ( k  e.  RR  ->  (
k  +  1 )  e.  RR )
2824, 27syl 16 . . . . . . . . . 10  |-  ( k  e.  NN  ->  (
k  +  1 )  e.  RR )
29 peano2nn 10004 . . . . . . . . . . 11  |-  ( k  e.  NN  ->  (
k  +  1 )  e.  NN )
3029nngt0d 10035 . . . . . . . . . 10  |-  ( k  e.  NN  ->  0  <  ( k  +  1 ) )
31 lerec 9884 . . . . . . . . . 10  |-  ( ( ( k  e.  RR  /\  0  <  k )  /\  ( ( k  +  1 )  e.  RR  /\  0  < 
( k  +  1 ) ) )  -> 
( k  <_  (
k  +  1 )  <-> 
( 1  /  (
k  +  1 ) )  <_  ( 1  /  k ) ) )
3224, 26, 28, 30, 31syl22anc 1185 . . . . . . . . 9  |-  ( k  e.  NN  ->  (
k  <_  ( k  +  1 )  <->  ( 1  /  ( k  +  1 ) )  <_ 
( 1  /  k
) ) )
3325, 32mpbid 202 . . . . . . . 8  |-  ( k  e.  NN  ->  (
1  /  ( k  +  1 ) )  <_  ( 1  / 
k ) )
34 oveq2 6081 . . . . . . . . . 10  |-  ( n  =  ( k  +  1 )  ->  (
1  /  n )  =  ( 1  / 
( k  +  1 ) ) )
35 ovex 6098 . . . . . . . . . 10  |-  ( 1  /  ( k  +  1 ) )  e. 
_V
3634, 13, 35fvmpt 5798 . . . . . . . . 9  |-  ( ( k  +  1 )  e.  NN  ->  ( F `  ( k  +  1 ) )  =  ( 1  / 
( k  +  1 ) ) )
3729, 36syl 16 . . . . . . . 8  |-  ( k  e.  NN  ->  ( F `  ( k  +  1 ) )  =  ( 1  / 
( k  +  1 ) ) )
3833, 37, 153brtr4d 4234 . . . . . . 7  |-  ( k  e.  NN  ->  ( F `  ( k  +  1 ) )  <_  ( F `  k ) )
3938adantl 453 . . . . . 6  |-  ( ( H  e.  dom  ~~>  /\  k  e.  NN )  ->  ( F `  ( k  +  1 ) )  <_  ( F `  k ) )
40 oveq2 6081 . . . . . . . . 9  |-  ( k  =  j  ->  (
2 ^ k )  =  ( 2 ^ j ) )
4140fveq2d 5724 . . . . . . . . 9  |-  ( k  =  j  ->  ( F `  ( 2 ^ k ) )  =  ( F `  ( 2 ^ j
) ) )
4240, 41oveq12d 6091 . . . . . . . 8  |-  ( k  =  j  ->  (
( 2 ^ k
)  x.  ( F `
 ( 2 ^ k ) ) )  =  ( ( 2 ^ j )  x.  ( F `  (
2 ^ j ) ) ) )
43 fconstmpt 4913 . . . . . . . . 9  |-  ( NN0 
X.  { 1 } )  =  ( k  e.  NN0  |->  1 )
44 2nn 10125 . . . . . . . . . . . . . 14  |-  2  e.  NN
45 nnexpcl 11386 . . . . . . . . . . . . . 14  |-  ( ( 2  e.  NN  /\  k  e.  NN0 )  -> 
( 2 ^ k
)  e.  NN )
4644, 45mpan 652 . . . . . . . . . . . . 13  |-  ( k  e.  NN0  ->  ( 2 ^ k )  e.  NN )
47 oveq2 6081 . . . . . . . . . . . . . 14  |-  ( n  =  ( 2 ^ k )  ->  (
1  /  n )  =  ( 1  / 
( 2 ^ k
) ) )
48 ovex 6098 . . . . . . . . . . . . . 14  |-  ( 1  /  ( 2 ^ k ) )  e. 
_V
4947, 13, 48fvmpt 5798 . . . . . . . . . . . . 13  |-  ( ( 2 ^ k )  e.  NN  ->  ( F `  ( 2 ^ k ) )  =  ( 1  / 
( 2 ^ k
) ) )
5046, 49syl 16 . . . . . . . . . . . 12  |-  ( k  e.  NN0  ->  ( F `
 ( 2 ^ k ) )  =  ( 1  /  (
2 ^ k ) ) )
5150oveq2d 6089 . . . . . . . . . . 11  |-  ( k  e.  NN0  ->  ( ( 2 ^ k )  x.  ( F `  ( 2 ^ k
) ) )  =  ( ( 2 ^ k )  x.  (
1  /  ( 2 ^ k ) ) ) )
52 nncn 10000 . . . . . . . . . . . . 13  |-  ( ( 2 ^ k )  e.  NN  ->  (
2 ^ k )  e.  CC )
53 nnne0 10024 . . . . . . . . . . . . 13  |-  ( ( 2 ^ k )  e.  NN  ->  (
2 ^ k )  =/=  0 )
5452, 53recidd 9777 . . . . . . . . . . . 12  |-  ( ( 2 ^ k )  e.  NN  ->  (
( 2 ^ k
)  x.  ( 1  /  ( 2 ^ k ) ) )  =  1 )
5546, 54syl 16 . . . . . . . . . . 11  |-  ( k  e.  NN0  ->  ( ( 2 ^ k )  x.  ( 1  / 
( 2 ^ k
) ) )  =  1 )
5651, 55eqtrd 2467 . . . . . . . . . 10  |-  ( k  e.  NN0  ->  ( ( 2 ^ k )  x.  ( F `  ( 2 ^ k
) ) )  =  1 )
5756mpteq2ia 4283 . . . . . . . . 9  |-  ( k  e.  NN0  |->  ( ( 2 ^ k )  x.  ( F `  ( 2 ^ k
) ) ) )  =  ( k  e. 
NN0  |->  1 )
5843, 57eqtr4i 2458 . . . . . . . 8  |-  ( NN0 
X.  { 1 } )  =  ( k  e.  NN0  |->  ( ( 2 ^ k )  x.  ( F `  ( 2 ^ k
) ) ) )
59 ovex 6098 . . . . . . . 8  |-  ( ( 2 ^ j )  x.  ( F `  ( 2 ^ j
) ) )  e. 
_V
6042, 58, 59fvmpt 5798 . . . . . . 7  |-  ( j  e.  NN0  ->  ( ( NN0  X.  { 1 } ) `  j
)  =  ( ( 2 ^ j )  x.  ( F `  ( 2 ^ j
) ) ) )
6160adantl 453 . . . . . 6  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN0 )  ->  (
( NN0  X.  { 1 } ) `  j
)  =  ( ( 2 ^ j )  x.  ( F `  ( 2 ^ j
) ) ) )
6218, 23, 39, 61climcnds 12623 . . . . 5  |-  ( H  e.  dom  ~~>  ->  (  seq  1 (  +  ,  F )  e.  dom  ~~>  <->  seq  0 (  +  , 
( NN0  X.  { 1 } ) )  e. 
dom 
~~>  ) )
6311, 62mpbid 202 . . . 4  |-  ( H  e.  dom  ~~>  ->  seq  0 (  +  , 
( NN0  X.  { 1 } ) )  e. 
dom 
~~>  )
641, 3, 6, 8, 63isumrecl 12541 . . 3  |-  ( H  e.  dom  ~~>  ->  sum_ k  e.  NN0  1  e.  RR )
65 arch 10210 . . 3  |-  ( sum_ k  e.  NN0  1  e.  RR  ->  E. j  e.  NN  sum_ k  e.  NN0  1  <  j )
6664, 65syl 16 . 2  |-  ( H  e.  dom  ~~>  ->  E. j  e.  NN  sum_ k  e.  NN0  1  <  j )
67 fzfid 11304 . . . . . . 7  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  (
1 ... j )  e. 
Fin )
68 ax-1cn 9040 . . . . . . 7  |-  1  e.  CC
69 fsumconst 12565 . . . . . . 7  |-  ( ( ( 1 ... j
)  e.  Fin  /\  1  e.  CC )  -> 
sum_ k  e.  ( 1 ... j ) 1  =  ( (
# `  ( 1 ... j ) )  x.  1 ) )
7067, 68, 69sylancl 644 . . . . . 6  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  sum_ k  e.  ( 1 ... j
) 1  =  ( ( # `  (
1 ... j ) )  x.  1 ) )
71 nnnn0 10220 . . . . . . . . 9  |-  ( j  e.  NN  ->  j  e.  NN0 )
7271adantl 453 . . . . . . . 8  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  j  e.  NN0 )
73 hashfz1 11622 . . . . . . . 8  |-  ( j  e.  NN0  ->  ( # `  ( 1 ... j
) )  =  j )
7472, 73syl 16 . . . . . . 7  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  ( # `
 ( 1 ... j ) )  =  j )
7574oveq1d 6088 . . . . . 6  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  (
( # `  ( 1 ... j ) )  x.  1 )  =  ( j  x.  1 ) )
76 nncn 10000 . . . . . . . 8  |-  ( j  e.  NN  ->  j  e.  CC )
7776adantl 453 . . . . . . 7  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  j  e.  CC )
7877mulid1d 9097 . . . . . 6  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  (
j  x.  1 )  =  j )
7970, 75, 783eqtrd 2471 . . . . 5  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  sum_ k  e.  ( 1 ... j
) 1  =  j )
802a1i 11 . . . . . 6  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  0  e.  ZZ )
81 elfznn 11072 . . . . . . . . 9  |-  ( k  e.  ( 1 ... j )  ->  k  e.  NN )
82 nnnn0 10220 . . . . . . . . 9  |-  ( k  e.  NN  ->  k  e.  NN0 )
8381, 82syl 16 . . . . . . . 8  |-  ( k  e.  ( 1 ... j )  ->  k  e.  NN0 )
8483ssriv 3344 . . . . . . 7  |-  ( 1 ... j )  C_  NN0
8584a1i 11 . . . . . 6  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  (
1 ... j )  C_  NN0 )
865adantl 453 . . . . . 6  |-  ( ( ( H  e.  dom  ~~>  /\  j  e.  NN )  /\  k  e.  NN0 )  ->  ( ( NN0 
X.  { 1 } ) `  k )  =  1 )
877a1i 11 . . . . . 6  |-  ( ( ( H  e.  dom  ~~>  /\  j  e.  NN )  /\  k  e.  NN0 )  ->  1  e.  RR )
88 0le1 9543 . . . . . . 7  |-  0  <_  1
8988a1i 11 . . . . . 6  |-  ( ( ( H  e.  dom  ~~>  /\  j  e.  NN )  /\  k  e.  NN0 )  ->  0  <_  1
)
9063adantr 452 . . . . . 6  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  seq  0 (  +  , 
( NN0  X.  { 1 } ) )  e. 
dom 
~~>  )
911, 80, 67, 85, 86, 87, 89, 90isumless 12617 . . . . 5  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  sum_ k  e.  ( 1 ... j
) 1  <_  sum_ k  e.  NN0  1 )
9279, 91eqbrtrrd 4226 . . . 4  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  j  <_ 
sum_ k  e.  NN0  1 )
93 nnre 9999 . . . . 5  |-  ( j  e.  NN  ->  j  e.  RR )
94 lenlt 9146 . . . . 5  |-  ( ( j  e.  RR  /\  sum_ k  e.  NN0  1  e.  RR )  ->  (
j  <_  sum_ k  e. 
NN0  1  <->  -.  sum_ k  e.  NN0  1  <  j
) )
9593, 64, 94syl2anr 465 . . . 4  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  (
j  <_  sum_ k  e. 
NN0  1  <->  -.  sum_ k  e.  NN0  1  <  j
) )
9692, 95mpbid 202 . . 3  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  -.  sum_ k  e.  NN0  1  <  j )
9796nrexdv 2801 . 2  |-  ( H  e.  dom  ~~>  ->  -.  E. j  e.  NN  sum_ k  e.  NN0  1  < 
j )
9866, 97pm2.65i 167 1  |-  -.  H  e.  dom  ~~>
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   E.wrex 2698    C_ wss 3312   {csn 3806   class class class wbr 4204    e. cmpt 4258    X. cxp 4868   dom cdm 4870   ` cfv 5446  (class class class)co 6073   Fincfn 7101   CCcc 8980   RRcr 8981   0cc0 8982   1c1 8983    + caddc 8985    x. cmul 8987    < clt 9112    <_ cle 9113    / cdiv 9669   NNcn 9992   2c2 10041   NN0cn0 10213   ZZcz 10274   ...cfz 11035    seq cseq 11315   ^cexp 11374   #chash 11610    ~~> cli 12270   sum_csu 12471
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-pm 7013  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7471  df-card 7818  df-cda 8040  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-ico 10914  df-fz 11036  df-fzo 11128  df-fl 11194  df-seq 11316  df-exp 11375  df-hash 11611  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-clim 12274  df-rlim 12275  df-sum 12472
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