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Mirrors > Home > MPE Home > Th. List > divcnv | Structured version Visualization version GIF version |
Description: The sequence of reciprocals of positive integers, multiplied by the factor 𝐴, converges to zero. (Contributed by NM, 6-Feb-2008.) (Revised by Mario Carneiro, 18-Sep-2014.) |
Ref | Expression |
---|---|
divcnv | ⊢ (𝐴 ∈ ℂ → (𝑛 ∈ ℕ ↦ (𝐴 / 𝑛)) ⇝ 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnrp 12388 | . . . . 5 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ+) | |
2 | 1 | ssriv 3968 | . . . 4 ⊢ ℕ ⊆ ℝ+ |
3 | 2 | a1i 11 | . . 3 ⊢ (𝐴 ∈ ℂ → ℕ ⊆ ℝ+) |
4 | divrcnv 15195 | . . 3 ⊢ (𝐴 ∈ ℂ → (𝑛 ∈ ℝ+ ↦ (𝐴 / 𝑛)) ⇝𝑟 0) | |
5 | 3, 4 | rlimres2 14906 | . 2 ⊢ (𝐴 ∈ ℂ → (𝑛 ∈ ℕ ↦ (𝐴 / 𝑛)) ⇝𝑟 0) |
6 | nnuz 12269 | . . 3 ⊢ ℕ = (ℤ≥‘1) | |
7 | 1zzd 12001 | . . 3 ⊢ (𝐴 ∈ ℂ → 1 ∈ ℤ) | |
8 | simpl 483 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ ℂ) | |
9 | nncn 11634 | . . . . . 6 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℂ) | |
10 | 9 | adantl 482 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℂ) |
11 | nnne0 11659 | . . . . . 6 ⊢ (𝑛 ∈ ℕ → 𝑛 ≠ 0) | |
12 | 11 | adantl 482 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑛 ∈ ℕ) → 𝑛 ≠ 0) |
13 | 8, 10, 12 | divcld 11404 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑛 ∈ ℕ) → (𝐴 / 𝑛) ∈ ℂ) |
14 | 13 | fmpttd 6871 | . . 3 ⊢ (𝐴 ∈ ℂ → (𝑛 ∈ ℕ ↦ (𝐴 / 𝑛)):ℕ⟶ℂ) |
15 | 6, 7, 14 | rlimclim 14891 | . 2 ⊢ (𝐴 ∈ ℂ → ((𝑛 ∈ ℕ ↦ (𝐴 / 𝑛)) ⇝𝑟 0 ↔ (𝑛 ∈ ℕ ↦ (𝐴 / 𝑛)) ⇝ 0)) |
16 | 5, 15 | mpbid 233 | 1 ⊢ (𝐴 ∈ ℂ → (𝑛 ∈ ℕ ↦ (𝐴 / 𝑛)) ⇝ 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2105 ≠ wne 3013 ⊆ wss 3933 class class class wbr 5057 ↦ cmpt 5137 (class class class)co 7145 ℂcc 10523 0cc0 10525 1c1 10526 / cdiv 11285 ℕcn 11626 ℝ+crp 12377 ⇝ cli 14829 ⇝𝑟 crli 14830 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-pm 8398 df-en 8498 df-dom 8499 df-sdom 8500 df-sup 8894 df-inf 8895 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-fl 13150 df-seq 13358 df-exp 13418 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-clim 14833 df-rlim 14834 |
This theorem is referenced by: divcnvshft 15198 supcvg 15199 expcnv 15207 plyeq0lem 24727 leibpi 25447 emcllem4 25503 basellem6 25590 circum 32814 divcnvlin 32861 hashnzfzclim 40531 clim1fr1 41758 divcnvg 41784 fprodsubrecnncnvlem 42067 fprodaddrecnncnvlem 42069 stirlinglem1 42236 |
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