Volume 6, 1994

University of Bialystok

Copyright (c) 1994 Association of Mizar Users

### The abstract of the Mizar article:

### Sequences in $\calE^N_\rmT$

**by****Agnieszka Sakowicz,****Jaroslaw Gryko, and****Adam Grabowski**- Received May 10, 1994
- MML identifier: TOPRNS_1

- [ Mizar article, MML identifier index ]

environ vocabulary FUNCT_1, PRE_TOPC, EUCLID, SEQ_1, RELAT_1, ANPROJ_1, BOOLE, ARYTM_1, COMPLEX1, FINSEQ_1, ABSVALUE, LATTICES, SEQ_2, ORDINAL2, ARYTM_3, NORMSP_1; notation TARSKI, XBOOLE_0, NUMBERS, XREAL_0, RELAT_1, FUNCT_1, REAL_1, NAT_1, SEQ_1, ABSVALUE, STRUCT_0, NORMSP_1, FINSEQ_1, FINSEQ_2, EUCLID, PRE_TOPC; constructors SEQ_1, ABSVALUE, EUCLID, REAL_1, NAT_1, PARTFUN1, NORMSP_1, MEMBERED, XBOOLE_0; clusters STRUCT_0, EUCLID, XREAL_0, RELSET_1, SEQ_1, ARYTM_3, MEMBERED, ZFMISC_1, XBOOLE_0, NUMBERS, ORDINAL2; requirements REAL, NUMERALS, SUBSET, BOOLE, ARITHM; begin definition let N be Nat; mode Real_Sequence of N is sequence of TOP-REAL N; end; reserve N,n,m,k,n1,n2 for Nat; reserve q,r,r1,r2 for Real; reserve x,y for set; reserve w,w1,w2,g,g1,g2 for Point of TOP-REAL N; reserve seq,seq1,seq2,seq3,seq' for Real_Sequence of N; canceled; theorem :: TOPRNS_1:2 for f being Function holds f is Real_Sequence of N iff (dom f=NAT & for n holds f.n is Point of TOP-REAL N); definition let N; let IT be Real_Sequence of N; attr IT is non-zero means :: TOPRNS_1:def 1 rng IT c= (the carrier of TOP-REAL N) \ {0.REAL N}; end; theorem :: TOPRNS_1:3 seq is non-zero iff for x st x in NAT holds seq.x<>0.REAL N; theorem :: TOPRNS_1:4 seq is non-zero iff for n holds seq.n<>0.REAL N; theorem :: TOPRNS_1:5 for N,seq,seq1 st (for x st x in NAT holds seq.x=seq1.x) holds seq=seq1; theorem :: TOPRNS_1:6 for N,seq,seq1 st (for n holds seq.n=seq1.n) holds seq=seq1; scheme ExTopRealNSeq{N()->Nat, F(Nat)->Point of TOP-REAL N()}: ex seq being Real_Sequence of N() st for n holds seq.n=F(n); definition let N,seq1,seq2; func seq1 + seq2 -> Real_Sequence of N means :: TOPRNS_1:def 2 for n holds it.n = seq1.n + seq2.n; end; definition let r,N,seq; func r*seq -> Real_Sequence of N means :: TOPRNS_1:def 3 for n holds it.n = r*seq.n; end; definition let N,seq; func - seq -> Real_Sequence of N means :: TOPRNS_1:def 4 for n holds it.n = -seq.n; end; definition let N,seq1,seq2; func seq1-seq2 -> Real_Sequence of N equals :: TOPRNS_1:def 5 seq1 +- seq2; end; definition let N; let x be Point of TOP-REAL N; func |.x.|-> Real means :: TOPRNS_1:def 6 ex y be FinSequence of REAL st x=y & it=|.y.|; end; definition let N,seq; func |.seq.| -> Real_Sequence means :: TOPRNS_1:def 7 for n holds it.n=|.seq.n.|; end; canceled; theorem :: TOPRNS_1:8 abs(r)*|.w.| = |.r*w.|; theorem :: TOPRNS_1:9 |.r*seq.| = abs(r)(#)|.seq.|; theorem :: TOPRNS_1:10 seq1 + seq2 = seq2 + seq1; theorem :: TOPRNS_1:11 (seq1 + seq2) + seq3 = seq1 + (seq2 + seq3); theorem :: TOPRNS_1:12 -seq = (-1)*seq; theorem :: TOPRNS_1:13 r*(seq1 + seq2) = r*seq1 + r*seq2; theorem :: TOPRNS_1:14 (r*q)*seq = r*(q*seq); theorem :: TOPRNS_1:15 r*(seq1 - seq2) = r*seq1 - r*seq2; theorem :: TOPRNS_1:16 seq1 - (seq2 + seq3) = seq1 - seq2 - seq3; theorem :: TOPRNS_1:17 1*seq=seq; theorem :: TOPRNS_1:18 -(-seq) = seq; theorem :: TOPRNS_1:19 seq1 - (-seq2) = seq1 + seq2; theorem :: TOPRNS_1:20 seq1 - (seq2 - seq3) = seq1 - seq2 + seq3; theorem :: TOPRNS_1:21 seq1 + (seq2 - seq3) = seq1 + seq2 - seq3; theorem :: TOPRNS_1:22 r <> 0 & seq is non-zero implies r*seq is non-zero; theorem :: TOPRNS_1:23 seq is non-zero implies -seq is non-zero; theorem :: TOPRNS_1:24 |.0.REAL N.| = 0; theorem :: TOPRNS_1:25 |.w.| = 0 implies w = 0.REAL N; theorem :: TOPRNS_1:26 |.w.| >= 0; theorem :: TOPRNS_1:27 |.-w.| = |.w.|; theorem :: TOPRNS_1:28 |.w1 - w2.| = |.w2 - w1.|; theorem :: TOPRNS_1:29 |.w1 - w2.| = 0 iff w1 = w2; theorem :: TOPRNS_1:30 |.w1 + w2.| <= |.w1.| + |.w2.|; theorem :: TOPRNS_1:31 |.w1 - w2.| <= |.w1.| + |.w2.|; theorem :: TOPRNS_1:32 |.w1.| - |.w2.| <= |.w1 + w2.|; theorem :: TOPRNS_1:33 |.w1.| - |.w2.| <= |.w1 - w2.|; theorem :: TOPRNS_1:34 w1 <> w2 implies |.w1 - w2.| > 0; theorem :: TOPRNS_1:35 |.w1 - w2.| <= |.w1 - w.| + |.w - w2.|; theorem :: TOPRNS_1:36 0<=r1 & |.w1.|<|.w2.| & r1<r2 implies |.w1.|*r1<|.w2.|*r2; canceled; theorem :: TOPRNS_1:38 -|.w.|<r & r<|.w.| iff abs(r)<|.w.|; definition let N; let IT be Real_Sequence of N; attr IT is bounded means :: TOPRNS_1:def 8 ex r st for n holds |.IT.n.| < r; end; theorem :: TOPRNS_1:39 for n ex r st (0<r & for m st m<=n holds |.seq.m.| < r); definition let N; let IT be Real_Sequence of N; attr IT is convergent means :: TOPRNS_1:def 9 ex g st for r st 0<r ex n st for m st n<=m holds |.IT.m-g.| < r; end; definition let N,seq; assume seq is convergent; func lim seq -> Point of TOP-REAL N means :: TOPRNS_1:def 10 for r st 0<r ex n st for m st n<=m holds |.seq.m-it.| < r; end; canceled; theorem :: TOPRNS_1:41 seq is convergent & seq' is convergent implies seq + seq' is convergent; theorem :: TOPRNS_1:42 seq is convergent & seq' is convergent implies lim (seq + seq')=(lim seq)+(lim seq'); theorem :: TOPRNS_1:43 seq is convergent implies r*seq is convergent; theorem :: TOPRNS_1:44 seq is convergent implies lim(r*seq)=r*(lim seq); theorem :: TOPRNS_1:45 seq is convergent implies - seq is convergent; theorem :: TOPRNS_1:46 seq is convergent implies lim(-seq)=-(lim seq); theorem :: TOPRNS_1:47 seq is convergent & seq' is convergent implies seq - seq' is convergent; theorem :: TOPRNS_1:48 seq is convergent & seq' is convergent implies lim(seq - seq')=(lim seq)-(lim seq'); canceled; theorem :: TOPRNS_1:50 seq is convergent implies seq is bounded; theorem :: TOPRNS_1:51 seq is convergent implies ((lim seq) <> 0.REAL N implies ex n st for m st n<=m holds |.(lim seq).|/2 < |.seq.m.|);

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