Volume 1, 1989

University of Bialystok

Copyright (c) 1989 Association of Mizar Users

### The abstract of the Mizar article:

### Convergent Real Sequences. Upper and Lower Bound of Sets of Real Numbers

**by****Jaroslaw Kotowicz**- Received November 23, 1989
- MML identifier: SEQ_4

- [ Mizar article, MML identifier index ]

environ vocabulary ARYTM, SEQ_1, ORDINAL2, SEQM_3, ARYTM_3, RELAT_1, ARYTM_1, SEQ_2, LATTICES, ABSVALUE, FUNCT_1, PROB_1, SEQ_4, MEMBERED; notation TARSKI, XBOOLE_0, SUBSET_1, ORDINAL1, NUMBERS, XCMPLX_0, XREAL_0, REAL_1, RELAT_1, SEQ_1, SEQ_2, NAT_1, FUNCT_2, SEQM_3, ABSVALUE, MEMBERED; constructors REAL_1, SEQ_2, NAT_1, SEQM_3, ABSVALUE, PARTFUN1, XCMPLX_0, XREAL_0, MEMBERED, XBOOLE_0; clusters XREAL_0, RELSET_1, SEQ_1, SEQM_3, ARYTM_3, REAL_1, XBOOLE_0, NUMBERS, SUBSET_1, MEMBERED, ZFMISC_1, ORDINAL2; requirements REAL, NUMERALS, SUBSET, BOOLE, ARITHM; begin reserve n,k,k1,m,m1,n1,n2,l for Nat; reserve r,r1,r2,p,p1,g,g1,g2,s,s1,s2 for real number; reserve seq,seq1,seq2 for Real_Sequence; reserve Nseq for increasing Seq_of_Nat; reserve x for set; reserve X,Y for Subset of REAL; theorem :: SEQ_4:1 0<r1 & r1<=r & 0<=g implies g/r<=g/r1; canceled 2; theorem :: SEQ_4:4 0<s implies 0<s/3; canceled; theorem :: SEQ_4:6 0<g & 0<=r & g<=g1 & r<r1 implies g*r<g1*r1; theorem :: SEQ_4:7 0<=g & 0<=r & g<=g1 & r<=r1 implies g*r<=g1*r1; theorem :: SEQ_4:8 for X,Y st for r,p st r in X & p in Y holds r<p ex g st for r,p st r in X & p in Y holds r<=g & g<=p; theorem :: SEQ_4:9 :: ARCHIMEDES LAW 0<p & (ex r st r in X) & (for r st r in X holds r+p in X) implies for g ex r st r in X & g<r; theorem :: SEQ_4:10 for r ex n st r<n; definition let X be real-membered set; attr X is bounded_above means :: SEQ_4:def 1 ex p st for r st r in X holds r<=p; attr X is bounded_below means :: SEQ_4:def 2 ex p st for r st r in X holds p<=r; end; definition let X; attr X is bounded means :: SEQ_4:def 3 X is bounded_below bounded_above; end; canceled 3; theorem :: SEQ_4:14 X is bounded iff ex s st 0<s & for r st r in X holds abs(r)<s; definition let r; redefine func {r} -> Subset of REAL; end; theorem :: SEQ_4:15 {r} is bounded; theorem :: SEQ_4:16 for X being real-membered set holds X is non empty bounded_above implies ex g st (for r st r in X holds r<=g) & for s st 0<s ex r st r in X & g-s<r; theorem :: SEQ_4:17 for X being real-membered set holds (for r st r in X holds r<=g1) & (for s st 0<s ex r st (r in X & g1-s<r)) & (for r st r in X holds r<=g2) & (for s st 0<s ex r st (r in X & g2-s<r)) implies g1=g2; theorem :: SEQ_4:18 for X being real-membered set holds X is non empty bounded_below implies ex g st (for r st r in X holds g<=r) & (for s st 0<s ex r st r in X & r<g+s); theorem :: SEQ_4:19 for X being real-membered set holds (for r st r in X holds g1<=r) & (for s st 0<s ex r st (r in X & r<g1+s)) & (for r st r in X holds g2<=r) & (for s st 0<s ex r st (r in X & r<g2+s)) implies g1=g2; definition let X be real-membered set; assume X is non empty bounded_above; func upper_bound X -> real number means :: SEQ_4:def 4 (for r st r in X holds r<=it) & (for s st 0<s ex r st r in X & it-s<r); end; definition let X be real-membered set; assume X is non empty bounded_below; func lower_bound X -> real number means :: SEQ_4:def 5 (for r st r in X holds it<=r) & (for s st 0<s ex r st r in X & r<it+s); end; definition let X; redefine func upper_bound X -> Real; redefine func lower_bound X -> Real; end; canceled 2; theorem :: SEQ_4:22 lower_bound {r} = r & upper_bound {r} = r; theorem :: SEQ_4:23 lower_bound {r} = upper_bound {r}; theorem :: SEQ_4:24 X is bounded non empty implies lower_bound X <= upper_bound X; theorem :: SEQ_4:25 X is bounded non empty implies ((ex r,p st r in X & p in X & p<>r) iff lower_bound X < upper_bound X); :: :: Theorems about the Convergence and the Limit :: theorem :: SEQ_4:26 seq is convergent implies abs seq is convergent; theorem :: SEQ_4:27 seq is convergent implies lim abs seq = abs lim seq; theorem :: SEQ_4:28 abs seq is convergent & lim abs seq=0 implies seq is convergent & lim seq=0; theorem :: SEQ_4:29 seq1 is_subsequence_of seq & seq is convergent implies seq1 is convergent; theorem :: SEQ_4:30 seq1 is_subsequence_of seq & seq is convergent implies lim seq1=lim seq; theorem :: SEQ_4:31 seq is convergent & (ex k st for n st k<=n holds seq1.n=seq.n) implies seq1 is convergent; theorem :: SEQ_4:32 seq is convergent & (ex k st for n st k<=n holds seq1.n=seq.n) implies lim seq=lim seq1; theorem :: SEQ_4:33 seq is convergent implies ((seq ^\k) is convergent & lim (seq ^\k)=lim seq); canceled; theorem :: SEQ_4:35 seq is convergent & (ex k st seq=seq1 ^\k) implies seq1 is convergent; theorem :: SEQ_4:36 seq is convergent & (ex k st seq=seq1 ^\k) implies lim seq1 =lim seq; theorem :: SEQ_4:37 seq is convergent & lim seq<>0 implies ex k st (seq ^\k) is_not_0; theorem :: SEQ_4:38 seq is convergent & lim seq<>0 implies ex seq1 st seq1 is_subsequence_of seq & seq1 is_not_0; theorem :: SEQ_4:39 seq is constant implies seq is convergent; theorem :: SEQ_4:40 (seq is constant & r in rng seq or seq is constant & (ex n st seq.n=r)) implies lim seq=r; theorem :: SEQ_4:41 seq is constant implies for n holds lim seq=seq.n; theorem :: SEQ_4:42 seq is convergent & lim seq<>0 implies for seq1 st seq1 is_subsequence_of seq & seq1 is_not_0 holds lim (seq1")=(lim seq)"; theorem :: SEQ_4:43 0<r & (for n holds seq.n=1/(n+r)) implies seq is convergent; theorem :: SEQ_4:44 0<r & (for n holds seq.n=1/(n+r)) implies lim seq=0; theorem :: SEQ_4:45 (for n holds seq.n=1/(n+1)) implies seq is convergent & lim seq=0; theorem :: SEQ_4:46 0<r & (for n holds seq.n=g/(n+r)) implies seq is convergent & lim seq=0; theorem :: SEQ_4:47 0<r & (for n holds seq.n=1/(n*n+r)) implies seq is convergent; theorem :: SEQ_4:48 0<r & (for n holds seq.n=1/(n*n+r)) implies lim seq=0; theorem :: SEQ_4:49 (for n holds seq.n=1/(n*n+1)) implies seq is convergent & lim seq=0; theorem :: SEQ_4:50 0<r & (for n holds seq.n=g/(n*n+r)) implies seq is convergent & lim seq=0; theorem :: SEQ_4:51 seq is non-decreasing & seq is bounded_above implies seq is convergent; theorem :: SEQ_4:52 seq is non-increasing & seq is bounded_below implies seq is convergent; theorem :: SEQ_4:53 seq is monotone & seq is bounded implies seq is convergent; theorem :: SEQ_4:54 seq is bounded_above & seq is non-decreasing implies for n holds seq.n<=lim seq; theorem :: SEQ_4:55 seq is bounded_below & seq is non-increasing implies for n holds lim seq <= seq.n; theorem :: SEQ_4:56 for seq ex Nseq st seq*Nseq is monotone; theorem :: SEQ_4:57 :: BOLZANO-WEIERSTRASS THEOREM seq is bounded implies ex seq1 st seq1 is_subsequence_of seq & seq1 is convergent; theorem :: SEQ_4:58 :: CAUCHY THEOREM seq is convergent iff for s st 0<s ex n st for m st n<=m holds abs(seq.m -seq.n)<s; theorem :: SEQ_4:59 seq is constant & seq1 is convergent implies lim (seq+seq1) =(seq.0) + lim seq1 & lim (seq-seq1) =(seq.0) - lim seq1 & lim (seq1-seq) =(lim seq1) -seq.0 & lim (seq(#)seq1) =(seq.0) * (lim seq1);

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