MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  harmonic Unicode version

Theorem harmonic 12191
Description: The harmonic series  H diverges. This fact follows from the stronger emcl 20128, 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
StepHypRef Expression
1 nn0uz 10141 . . . 4  |-  NN0  =  ( ZZ>= `  0 )
2 0z 9914 . . . . 5  |-  0  e.  ZZ
32a1i 12 . . . 4  |-  ( H  e.  dom  ~~>  ->  0  e.  ZZ )
4 1ex 8713 . . . . . 6  |-  1  e.  _V
54fvconst2 5581 . . . . 5  |-  ( k  e.  NN0  ->  ( ( NN0  X.  { 1 } ) `  k
)  =  1 )
65adantl 454 . . . 4  |-  ( ( H  e.  dom  ~~>  /\  k  e.  NN0 )  ->  (
( NN0  X.  { 1 } ) `  k
)  =  1 )
7 1re 8717 . . . . 5  |-  1  e.  RR
87a1i 12 . . . 4  |-  ( ( H  e.  dom  ~~>  /\  k  e.  NN0 )  ->  1  e.  RR )
9 harmonic.2 . . . . . . 7  |-  H  =  seq  1 (  +  ,  F )
109eleq1i 2316 . . . . . 6  |-  ( H  e.  dom  ~~>  <->  seq  1
(  +  ,  F
)  e.  dom  ~~>  )
1110biimpi 188 . . . . 5  |-  ( H  e.  dom  ~~>  ->  seq  1 (  +  ,  F )  e.  dom  ~~>  )
12 oveq2 5718 . . . . . . . . 9  |-  ( n  =  k  ->  (
1  /  n )  =  ( 1  / 
k ) )
13 harmonic.1 . . . . . . . . 9  |-  F  =  ( n  e.  NN  |->  ( 1  /  n
) )
14 ovex 5735 . . . . . . . . 9  |-  ( 1  /  k )  e. 
_V
1512, 13, 14fvmpt 5454 . . . . . . . 8  |-  ( k  e.  NN  ->  ( F `  k )  =  ( 1  / 
k ) )
16 nnrecre 9662 . . . . . . . 8  |-  ( k  e.  NN  ->  (
1  /  k )  e.  RR )
1715, 16eqeltrd 2327 . . . . . . 7  |-  ( k  e.  NN  ->  ( F `  k )  e.  RR )
1817adantl 454 . . . . . 6  |-  ( ( H  e.  dom  ~~>  /\  k  e.  NN )  ->  ( F `  k )  e.  RR )
19 nnrp 10242 . . . . . . . . . 10  |-  ( k  e.  NN  ->  k  e.  RR+ )
2019rpreccld 10279 . . . . . . . . 9  |-  ( k  e.  NN  ->  (
1  /  k )  e.  RR+ )
2120rpge0d 10273 . . . . . . . 8  |-  ( k  e.  NN  ->  0  <_  ( 1  /  k
) )
2221, 15breqtrrd 3946 . . . . . . 7  |-  ( k  e.  NN  ->  0  <_  ( F `  k
) )
2322adantl 454 . . . . . 6  |-  ( ( H  e.  dom  ~~>  /\  k  e.  NN )  ->  0  <_  ( F `  k
) )
24 nnre 9633 . . . . . . . . . 10  |-  ( k  e.  NN  ->  k  e.  RR )
2524lep1d 9568 . . . . . . . . 9  |-  ( k  e.  NN  ->  k  <_  ( k  +  1 ) )
26 nngt0 9655 . . . . . . . . . 10  |-  ( k  e.  NN  ->  0  <  k )
27 peano2re 8865 . . . . . . . . . . 11  |-  ( k  e.  RR  ->  (
k  +  1 )  e.  RR )
2824, 27syl 17 . . . . . . . . . 10  |-  ( k  e.  NN  ->  (
k  +  1 )  e.  RR )
29 peano2nn 9638 . . . . . . . . . . 11  |-  ( k  e.  NN  ->  (
k  +  1 )  e.  NN )
3029nngt0d 9669 . . . . . . . . . 10  |-  ( k  e.  NN  ->  0  <  ( k  +  1 ) )
31 lerec 9518 . . . . . . . . . 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 1188 . . . . . . . . 9  |-  ( k  e.  NN  ->  (
k  <_  ( k  +  1 )  <->  ( 1  /  ( k  +  1 ) )  <_ 
( 1  /  k
) ) )
3325, 32mpbid 203 . . . . . . . 8  |-  ( k  e.  NN  ->  (
1  /  ( k  +  1 ) )  <_  ( 1  / 
k ) )
34 oveq2 5718 . . . . . . . . . 10  |-  ( n  =  ( k  +  1 )  ->  (
1  /  n )  =  ( 1  / 
( k  +  1 ) ) )
35 ovex 5735 . . . . . . . . . 10  |-  ( 1  /  ( k  +  1 ) )  e. 
_V
3634, 13, 35fvmpt 5454 . . . . . . . . 9  |-  ( ( k  +  1 )  e.  NN  ->  ( F `  ( k  +  1 ) )  =  ( 1  / 
( k  +  1 ) ) )
3729, 36syl 17 . . . . . . . 8  |-  ( k  e.  NN  ->  ( F `  ( k  +  1 ) )  =  ( 1  / 
( k  +  1 ) ) )
3833, 37, 153brtr4d 3950 . . . . . . 7  |-  ( k  e.  NN  ->  ( F `  ( k  +  1 ) )  <_  ( F `  k ) )
3938adantl 454 . . . . . 6  |-  ( ( H  e.  dom  ~~>  /\  k  e.  NN )  ->  ( F `  ( k  +  1 ) )  <_  ( F `  k ) )
40 oveq2 5718 . . . . . . . . 9  |-  ( k  =  j  ->  (
2 ^ k )  =  ( 2 ^ j ) )
4140fveq2d 5381 . . . . . . . . 9  |-  ( k  =  j  ->  ( F `  ( 2 ^ k ) )  =  ( F `  ( 2 ^ j
) ) )
4240, 41oveq12d 5728 . . . . . . . 8  |-  ( k  =  j  ->  (
( 2 ^ k
)  x.  ( F `
 ( 2 ^ k ) ) )  =  ( ( 2 ^ j )  x.  ( F `  (
2 ^ j ) ) ) )
43 fconstmpt 4639 . . . . . . . . 9  |-  ( NN0 
X.  { 1 } )  =  ( k  e.  NN0  |->  1 )
44 2nn 9756 . . . . . . . . . . . . . 14  |-  2  e.  NN
45 nnexpcl 10994 . . . . . . . . . . . . . 14  |-  ( ( 2  e.  NN  /\  k  e.  NN0 )  -> 
( 2 ^ k
)  e.  NN )
4644, 45mpan 654 . . . . . . . . . . . . 13  |-  ( k  e.  NN0  ->  ( 2 ^ k )  e.  NN )
47 oveq2 5718 . . . . . . . . . . . . . 14  |-  ( n  =  ( 2 ^ k )  ->  (
1  /  n )  =  ( 1  / 
( 2 ^ k
) ) )
48 ovex 5735 . . . . . . . . . . . . . 14  |-  ( 1  /  ( 2 ^ k ) )  e. 
_V
4947, 13, 48fvmpt 5454 . . . . . . . . . . . . 13  |-  ( ( 2 ^ k )  e.  NN  ->  ( F `  ( 2 ^ k ) )  =  ( 1  / 
( 2 ^ k
) ) )
5046, 49syl 17 . . . . . . . . . . . 12  |-  ( k  e.  NN0  ->  ( F `
 ( 2 ^ k ) )  =  ( 1  /  (
2 ^ k ) ) )
5150oveq2d 5726 . . . . . . . . . . 11  |-  ( k  e.  NN0  ->  ( ( 2 ^ k )  x.  ( F `  ( 2 ^ k
) ) )  =  ( ( 2 ^ k )  x.  (
1  /  ( 2 ^ k ) ) ) )
52 nncn 9634 . . . . . . . . . . . . 13  |-  ( ( 2 ^ k )  e.  NN  ->  (
2 ^ k )  e.  CC )
53 nnne0 9658 . . . . . . . . . . . . 13  |-  ( ( 2 ^ k )  e.  NN  ->  (
2 ^ k )  =/=  0 )
5452, 53recidd 9411 . . . . . . . . . . . 12  |-  ( ( 2 ^ k )  e.  NN  ->  (
( 2 ^ k
)  x.  ( 1  /  ( 2 ^ k ) ) )  =  1 )
5546, 54syl 17 . . . . . . . . . . 11  |-  ( k  e.  NN0  ->  ( ( 2 ^ k )  x.  ( 1  / 
( 2 ^ k
) ) )  =  1 )
5651, 55eqtrd 2285 . . . . . . . . . 10  |-  ( k  e.  NN0  ->  ( ( 2 ^ k )  x.  ( F `  ( 2 ^ k
) ) )  =  1 )
5756mpteq2ia 3999 . . . . . . . . 9  |-  ( k  e.  NN0  |->  ( ( 2 ^ k )  x.  ( F `  ( 2 ^ k
) ) ) )  =  ( k  e. 
NN0  |->  1 )
5843, 57eqtr4i 2276 . . . . . . . 8  |-  ( NN0 
X.  { 1 } )  =  ( k  e.  NN0  |->  ( ( 2 ^ k )  x.  ( F `  ( 2 ^ k
) ) ) )
59 ovex 5735 . . . . . . . 8  |-  ( ( 2 ^ j )  x.  ( F `  ( 2 ^ j
) ) )  e. 
_V
6042, 58, 59fvmpt 5454 . . . . . . 7  |-  ( j  e.  NN0  ->  ( ( NN0  X.  { 1 } ) `  j
)  =  ( ( 2 ^ j )  x.  ( F `  ( 2 ^ j
) ) ) )
6160adantl 454 . . . . . 6  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN0 )  ->  (
( NN0  X.  { 1 } ) `  j
)  =  ( ( 2 ^ j )  x.  ( F `  ( 2 ^ j
) ) ) )
6218, 23, 39, 61climcnds 12184 . . . . 5  |-  ( H  e.  dom  ~~>  ->  (  seq  1 (  +  ,  F )  e.  dom  ~~>  <->  seq  0 (  +  , 
( NN0  X.  { 1 } ) )  e. 
dom 
~~>  ) )
6311, 62mpbid 203 . . . 4  |-  ( H  e.  dom  ~~>  ->  seq  0 (  +  , 
( NN0  X.  { 1 } ) )  e. 
dom 
~~>  )
641, 3, 6, 8, 63isumrecl 12105 . . 3  |-  ( H  e.  dom  ~~>  ->  sum_ k  e.  NN0  1  e.  RR )
65 arch 9841 . . 3  |-  ( sum_ k  e.  NN0  1  e.  RR  ->  E. j  e.  NN  sum_ k  e.  NN0  1  <  j )
6664, 65syl 17 . 2  |-  ( H  e.  dom  ~~>  ->  E. j  e.  NN  sum_ k  e.  NN0  1  <  j )
67 fzfid 10913 . . . . . . 7  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  (
1 ... j )  e. 
Fin )
68 ax-1cn 8675 . . . . . . 7  |-  1  e.  CC
69 fsumconst 12129 . . . . . . 7  |-  ( ( ( 1 ... j
)  e.  Fin  /\  1  e.  CC )  -> 
sum_ k  e.  ( 1 ... j ) 1  =  ( (
# `  ( 1 ... j ) )  x.  1 ) )
7067, 68, 69sylancl 646 . . . . . 6  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  sum_ k  e.  ( 1 ... j
) 1  =  ( ( # `  (
1 ... j ) )  x.  1 ) )
71 nnnn0 9851 . . . . . . . . 9  |-  ( j  e.  NN  ->  j  e.  NN0 )
7271adantl 454 . . . . . . . 8  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  j  e.  NN0 )
73 hashfz1 11223 . . . . . . . 8  |-  ( j  e.  NN0  ->  ( # `  ( 1 ... j
) )  =  j )
7472, 73syl 17 . . . . . . 7  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  ( # `
 ( 1 ... j ) )  =  j )
7574oveq1d 5725 . . . . . 6  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  (
( # `  ( 1 ... j ) )  x.  1 )  =  ( j  x.  1 ) )
76 nncn 9634 . . . . . . . 8  |-  ( j  e.  NN  ->  j  e.  CC )
7776adantl 454 . . . . . . 7  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  j  e.  CC )
7877mulid1d 8732 . . . . . 6  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  (
j  x.  1 )  =  j )
7970, 75, 783eqtrd 2289 . . . . 5  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  sum_ k  e.  ( 1 ... j
) 1  =  j )
802a1i 12 . . . . . 6  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  0  e.  ZZ )
81 elfznn 10697 . . . . . . . . 9  |-  ( k  e.  ( 1 ... j )  ->  k  e.  NN )
82 nnnn0 9851 . . . . . . . . 9  |-  ( k  e.  NN  ->  k  e.  NN0 )
8381, 82syl 17 . . . . . . . 8  |-  ( k  e.  ( 1 ... j )  ->  k  e.  NN0 )
8483ssriv 3105 . . . . . . 7  |-  ( 1 ... j )  C_  NN0
8584a1i 12 . . . . . 6  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  (
1 ... j )  C_  NN0 )
865adantl 454 . . . . . 6  |-  ( ( ( H  e.  dom  ~~>  /\  j  e.  NN )  /\  k  e.  NN0 )  ->  ( ( NN0 
X.  { 1 } ) `  k )  =  1 )
877a1i 12 . . . . . 6  |-  ( ( ( H  e.  dom  ~~>  /\  j  e.  NN )  /\  k  e.  NN0 )  ->  1  e.  RR )
88 0le1 9177 . . . . . . 7  |-  0  <_  1
8988a1i 12 . . . . . 6  |-  ( ( ( H  e.  dom  ~~>  /\  j  e.  NN )  /\  k  e.  NN0 )  ->  0  <_  1
)
9063adantr 453 . . . . . 6  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  seq  0 (  +  , 
( NN0  X.  { 1 } ) )  e. 
dom 
~~>  )
911, 80, 67, 85, 86, 87, 89, 90isumless 12178 . . . . 5  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  sum_ k  e.  ( 1 ... j
) 1  <_  sum_ k  e.  NN0  1 )
9279, 91eqbrtrrd 3942 . . . 4  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  j  <_ 
sum_ k  e.  NN0  1 )
93 nnre 9633 . . . . 5  |-  ( j  e.  NN  ->  j  e.  RR )
94 lenlt 8781 . . . . 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 466 . . . 4  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  (
j  <_  sum_ k  e. 
NN0  1  <->  -.  sum_ k  e.  NN0  1  <  j
) )
9692, 95mpbid 203 . . 3  |-  ( ( H  e.  dom  ~~>  /\  j  e.  NN )  ->  -.  sum_ k  e.  NN0  1  <  j )
9796nrexdv 2608 . 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 5    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621   E.wrex 2510    C_ wss 3078   {csn 3544   class class class wbr 3920    e. cmpt 3974    X. cxp 4578   dom cdm 4580   ` cfv 4592  (class class class)co 5710   Fincfn 6749   CCcc 8615   RRcr 8616   0cc0 8617   1c1 8618    + caddc 8620    x. cmul 8622    < clt 8747    <_ cle 8748    / cdiv 9303   NNcn 9626   2c2 9675   NN0cn0 9844   ZZcz 9903   ...cfz 10660    seq cseq 10924   ^cexp 10982   #chash 11215    ~~> cli 11835   sum_csu 12035
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-inf2 7226  ax-cnex 8673  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-pre-mulgt0 8694  ax-pre-sup 8695
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-se 4246  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-isom 4609  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-riota 6190  df-recs 6274  df-rdg 6309  df-1o 6365  df-oadd 6369  df-er 6546  df-pm 6661  df-en 6750  df-dom 6751  df-sdom 6752  df-fin 6753  df-sup 7078  df-oi 7109  df-card 7456  df-cda 7678  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-div 9304  df-n 9627  df-2 9684  df-3 9685  df-n0 9845  df-z 9904  df-uz 10110  df-rp 10234  df-ico 10540  df-fz 10661  df-fzo 10749  df-fl 10803  df-seq 10925  df-exp 10983  df-hash 11216  df-cj 11461  df-re 11462  df-im 11463  df-sqr 11597  df-abs 11598  df-clim 11839  df-rlim 11840  df-sum 12036
  Copyright terms: Public domain W3C validator