Volume 1, 1989

University of Bialystok

Copyright (c) 1989 Association of Mizar Users

### The abstract of the Mizar article:

### Convergent Sequences and the Limit of Sequences

**by****Jaroslaw Kotowicz**- Received July 4, 1989
- MML identifier: SEQ_2

- [ Mizar article, MML identifier index ]

environ vocabulary ARYTM, SEQ_1, ARYTM_1, ARYTM_3, RELAT_1, ABSVALUE, FUNCT_1, LATTICES, ORDINAL2, SEQ_2; notation ORDINAL1, NUMBERS, XCMPLX_0, XREAL_0, REAL_1, ABSVALUE, NAT_1, RELAT_1, FUNCT_1, SEQ_1; constructors REAL_1, SEQ_1, ABSVALUE, NAT_1, PARTFUN1, MEMBERED, XBOOLE_0; clusters SEQ_1, XREAL_0, ARYTM_3, RELSET_1, MEMBERED, ZFMISC_1, XBOOLE_0, ORDINAL2; requirements REAL, NUMERALS, SUBSET, BOOLE, ARITHM; begin reserve n,n1,n2,k,m for Nat, r,r1,r2,p,g1,g2,g for real number, seq,seq',seq1 for Real_Sequence, y for set, f for real-yielding Function; canceled 2; theorem :: SEQ_2:3 0<g implies 0<g/2 & 0<g/4; theorem :: SEQ_2:4 0<g implies g/2<g; canceled; theorem :: SEQ_2:6 0<g & 0<p implies 0<g/p; theorem :: SEQ_2:7 0<=g & 0<=r & g<g1 & r<r1 implies g*r<g1*r1; canceled; theorem :: SEQ_2:9 -g<r & r<g iff abs(r)<g; theorem :: SEQ_2:10 0<r1 & r1<r & 0<g implies g/r<g/r1; theorem :: SEQ_2:11 g<>0 & r<>0 implies abs(g"-r")=abs(g-r)/(abs(g)*abs(r)); definition let f be real-yielding Function; attr f is bounded_above means :: SEQ_2:def 1 ex r st for y st y in dom f holds f.y<r; attr f is bounded_below means :: SEQ_2:def 2 ex r st for y st y in dom f holds r<f.y; end; definition let seq; redefine attr seq is bounded_above means :: SEQ_2:def 3 ex r st for n holds seq.n<r; redefine attr seq is bounded_below means :: SEQ_2:def 4 ex r st for n holds r<seq.n; end; definition let f; attr f is bounded means :: SEQ_2:def 5 f is bounded_above bounded_below; end; definition cluster bounded -> bounded_above bounded_below (real-yielding Function); cluster bounded_above bounded_below -> bounded (real-yielding Function); end; canceled 3; theorem :: SEQ_2:15 seq is bounded iff ex r st (0<r & for n holds abs(seq.n)<r); theorem :: SEQ_2:16 for n ex r st (0<r & for m st m<=n holds abs(seq.m)<r); :: :: CONVERGENT REAL SEQUENCES :: THE LIMIT OF SEQUENCES :: definition let seq; attr seq is convergent means :: SEQ_2:def 6 ex g st for p st 0<p ex n st for m st n<=m holds abs (seq.m-g) < p; end; definition let seq; assume seq is convergent; func lim seq -> real number means :: SEQ_2:def 7 for p st 0<p ex n st for m st n<=m holds abs(seq.m-it)<p; end; definition let seq; redefine func lim seq -> Real; end; canceled 2; theorem :: SEQ_2:19 seq is convergent & seq' is convergent implies seq + seq' is convergent; theorem :: SEQ_2:20 seq is convergent & seq' is convergent implies lim (seq + seq')=(lim seq)+(lim seq'); theorem :: SEQ_2:21 seq is convergent implies r(#)seq is convergent; theorem :: SEQ_2:22 seq is convergent implies lim(r(#)seq)=r*(lim seq); theorem :: SEQ_2:23 seq is convergent implies - seq is convergent; theorem :: SEQ_2:24 seq is convergent implies lim(-seq)=-(lim seq); theorem :: SEQ_2:25 seq is convergent & seq' is convergent implies seq - seq' is convergent; theorem :: SEQ_2:26 seq is convergent & seq' is convergent implies lim(seq - seq')=(lim seq)-(lim seq'); theorem :: SEQ_2:27 seq is convergent implies seq is bounded; theorem :: SEQ_2:28 seq is convergent & seq' is convergent implies seq (#) seq' is convergent; theorem :: SEQ_2:29 seq is convergent & seq' is convergent implies lim(seq(#)seq')=(lim seq)*(lim seq'); theorem :: SEQ_2:30 seq is convergent implies ((lim seq)<>0 implies ex n st for m st n<=m holds abs(lim seq)/2<abs(seq.m)); theorem :: SEQ_2:31 seq is convergent & (for n holds 0<=seq.n) implies 0<=(lim seq); theorem :: SEQ_2:32 seq is convergent & seq' is convergent & (for n holds seq.n<=(seq'.n)) implies (lim seq)<=(lim seq'); theorem :: SEQ_2:33 seq is convergent & seq' is convergent & (for n holds seq.n<=(seq1.n) & seq1.n<=seq'.n) & (lim seq)=(lim seq') implies seq1 is convergent; theorem :: SEQ_2:34 seq is convergent & seq' is convergent & (for n holds seq.n<=(seq1.n) & seq1.n<=seq'.n) & (lim seq)=(lim seq') implies (lim seq1)=(lim seq); theorem :: SEQ_2:35 seq is convergent & (lim seq)<>0 & seq is_not_0 implies seq" is convergent; theorem :: SEQ_2:36 seq is convergent & (lim seq)<>0 & seq is_not_0 implies lim seq"=(lim seq)"; theorem :: SEQ_2:37 seq' is convergent & seq is convergent & (lim seq)<>0 & seq is_not_0 implies seq'/"seq is convergent; theorem :: SEQ_2:38 seq' is convergent & seq is convergent & (lim seq)<>0 & seq is_not_0 implies lim(seq'/"seq)=(lim seq')/(lim seq); theorem :: SEQ_2:39 seq is convergent & seq1 is bounded & lim seq=0 implies seq(#)seq1 is convergent; theorem :: SEQ_2:40 seq is convergent & seq1 is bounded & lim seq=0 implies lim(seq(#)seq1)=0;

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