Subset of REAL;
end;
theorem :: SEQ_4:5
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~~ Real means
:: SEQ_4:def 1
(for r st r in X holds r<=it) & for s st 0~~~~ Real means
:: SEQ_4:def 2
(for r st r in X holds it<=r) & for s st 0~~~~r) iff
lower_bound X < upper_bound X);
::
:: Theorems about the Convergence and the Limit
::
theorem :: SEQ_4:13
seq is convergent implies abs seq is convergent;
registration
let seq be convergent Real_Sequence;
cluster abs seq -> convergent for Real_Sequence;
end;
theorem :: SEQ_4:14
seq is convergent implies lim abs seq = |.lim seq.|;
theorem :: SEQ_4:15
abs seq is convergent & lim abs seq=0 implies seq is convergent & lim seq=0;
theorem :: SEQ_4:16
seq1 is subsequence of seq & seq is convergent implies seq1 is convergent;
theorem :: SEQ_4:17
seq1 is subsequence of seq & seq is convergent implies lim seq1= lim seq;
theorem :: SEQ_4:18
seq is convergent & (ex k st for n st k<=n holds seq1.n=seq.n)
implies seq1 is convergent;
theorem :: SEQ_4:19
seq is convergent & (ex k st for n st k<=n holds seq1.n=seq.n) implies
lim seq=lim seq1;
registration
let seq be convergent Real_Sequence;
let k;
cluster seq ^\ k -> convergent for Real_Sequence;
end;
theorem :: SEQ_4:20
seq is convergent implies lim (seq^\k)=lim seq;
theorem :: SEQ_4:21
seq^\k is convergent implies seq is convergent;
theorem :: SEQ_4:22
seq^\k is convergent implies lim seq = lim (seq^\k);
theorem :: SEQ_4:23
seq is convergent & lim seq<>0 implies ex k st (seq ^\k) is non-zero;
theorem :: SEQ_4:24
seq is convergent & lim seq<>0 implies ex seq1 st seq1 is subsequence
of seq & seq1 is non-zero;
theorem :: SEQ_4:25
seq is constant & (r in rng seq or ex n st seq.n=r ) implies lim seq=r;
theorem :: SEQ_4:26
seq is constant implies for n holds lim seq=seq.n;
theorem :: SEQ_4:27
seq is convergent & lim seq<>0 implies for seq1 st seq1 is subsequence
of seq & seq1 is non-zero holds lim (seq1")=(lim seq)";
theorem :: SEQ_4:28
0<=r & (for n holds seq.n=1/(n+r)) implies seq is convergent;
theorem :: SEQ_4:29
0<=r & (for n holds seq.n=1/(n+r)) implies lim seq=0;
theorem :: SEQ_4:30
(for n holds seq.n=1/(n+1)) implies seq is convergent & lim seq=0;
theorem :: SEQ_4:31
0<=r & (for n holds seq.n=g/(n+r)) implies seq is convergent & lim seq =0;
theorem :: SEQ_4:32
0<=r & (for n holds seq.n=1/(n*n+r)) implies seq is convergent;
theorem :: SEQ_4:33
0<=r & (for n holds seq.n=1/(n*n+r)) implies lim seq=0;
theorem :: SEQ_4:34
(for n holds seq.n=1/(n*n+1)) implies seq is convergent & lim seq=0;
theorem :: SEQ_4:35
0<=r & (for n holds seq.n=g/(n*n+r)) implies seq is convergent & lim seq=0;
registration
cluster non-decreasing bounded_above -> convergent for Real_Sequence;
end;
registration
cluster non-increasing bounded_below -> convergent for Real_Sequence;
end;
registration
cluster monotone bounded -> convergent for Real_Sequence;
end;
theorem :: SEQ_4:36
seq is monotone & seq is bounded implies seq is convergent;
theorem :: SEQ_4:37
seq is bounded_above & seq is non-decreasing implies for n holds seq.n
<=lim seq;
theorem :: SEQ_4:38
seq is bounded_below & seq is non-increasing implies for n holds lim
seq <= seq.n;
theorem :: SEQ_4:39
for seq ex Nseq st seq*Nseq is monotone;
::$N Bolzano-Weierstrass Theorem (1 dimension)
theorem :: SEQ_4:40
seq is bounded implies ex seq1 st seq1 is subsequence of seq &
seq1 is convergent;
::$N Cauchy sequence
theorem :: SEQ_4:41 :: Cauchy Theorem
seq is convergent iff for s st 0~~~~= t holds lower_bound X >= t;
theorem :: SEQ_4:44
for X being non empty real-membered set st (for s st s in X holds s >=
r) & for t st for s st s in X holds s >= t holds r >= t holds r = lower_bound X
;
theorem :: SEQ_4:45
for X being non empty real-membered set, r for t st for s st s
in X holds s <= t holds upper_bound X <= t;
theorem :: SEQ_4:46
for X being non empty real-membered set, r st (for s st s in X holds s
<= r) & for t st for s st s in X holds s <= t holds r <= t holds r =
upper_bound X;
theorem :: SEQ_4:47
for X being non empty real-membered set, Y being real-membered set st
X c= Y & Y is bounded_below holds lower_bound Y <= lower_bound X;
theorem :: SEQ_4:48
for X being non empty real-membered set, Y being real-membered set st
X c= Y & Y is bounded_above holds upper_bound X <= upper_bound Y;
:: from CQC_SIM1, 2007.03.15, A.T.
definition
let A be non empty natural-membered set;
redefine func min A -> Element of NAT;
end;
begin :: moved from COMPLSP1, 2010.02.25
reserve k,n for Nat,
r,r9,r1,r2 for Real,
c,c9,c1,c2,c3 for Element of COMPLEX;
theorem :: SEQ_4:49
0c is_a_unity_wrt addcomplex;
theorem :: SEQ_4:50
compcomplex is_an_inverseOp_wrt addcomplex;
theorem :: SEQ_4:51
addcomplex is having_an_inverseOp;
theorem :: SEQ_4:52
the_inverseOp_wrt addcomplex = compcomplex;
definition
redefine func diffcomplex equals
:: SEQ_4:def 3
addcomplex*(id COMPLEX,compcomplex);
end;
theorem :: SEQ_4:53
1r is_a_unity_wrt multcomplex;
theorem :: SEQ_4:54
multcomplex is_distributive_wrt addcomplex;
definition let c be Complex;
func c multcomplex -> UnOp of COMPLEX equals
:: SEQ_4:def 4
multcomplex[;](c,id COMPLEX);
end;
theorem :: SEQ_4:55
(c multcomplex).c9 = c*c9;
theorem :: SEQ_4:56
c multcomplex is_distributive_wrt addcomplex;
definition
func abscomplex -> Function of COMPLEX,REAL means
:: SEQ_4:def 5
for c holds it.c = |.c.|;
end;
reserve z,z1,z2 for FinSequence of COMPLEX;
definition let z1,z2;
redefine func z1 + z2 -> FinSequence of COMPLEX equals
:: SEQ_4:def 6
addcomplex.:(z1,z2);
redefine func z1 - z2 -> FinSequence of COMPLEX equals
:: SEQ_4:def 7
diffcomplex.:(z1,z2);
end;
definition let z;
redefine func -z -> FinSequence of COMPLEX equals
:: SEQ_4:def 8
compcomplex*z;
end;
notation let z;
let c be Complex;
synonym c*z for c(#)z;
end;
definition let z;
let c be Complex;
redefine func c*z -> FinSequence of COMPLEX equals
:: SEQ_4:def 9
(c multcomplex)*z;
end;
definition let z;
redefine func abs z -> FinSequence of REAL equals
:: SEQ_4:def 10
abscomplex*z;
end;
definition let n;
func COMPLEX n -> FinSequenceSet of COMPLEX equals
:: SEQ_4:def 11
n-tuples_on COMPLEX;
end;
registration let n;
cluster COMPLEX n -> non empty;
end;
reserve x,z,z1,z2,z3 for Element of COMPLEX n,
A,B for Subset of COMPLEX n;
theorem :: SEQ_4:57
k in Seg n implies z.k in COMPLEX;
definition let n,z1,z2;
redefine func z1 + z2 -> Element of COMPLEX n;
end;
theorem :: SEQ_4:58
k in Seg n & c1 = z1.k & c2 = z2.k implies (z1 + z2).k = c1 + c2;
definition let n;
func 0c n -> FinSequence of COMPLEX equals
:: SEQ_4:def 12
n |-> 0c;
end;
definition let n;
redefine func 0c n -> Element of COMPLEX n;
end;
theorem :: SEQ_4:59
z + 0c n = z & z = 0c n + z;
definition let n,z;
redefine func -z -> Element of COMPLEX n;
end;
theorem :: SEQ_4:60
k in Seg n & c = z.k implies (-z).k = -c;
theorem :: SEQ_4:61
z + -z = 0c n & -z + z = 0c n;
theorem :: SEQ_4:62
z1 + z2 = 0c n implies z1 = -z2 & z2 = -z1
;
::$CT
theorem :: SEQ_4:64
-z1 = -z2 implies z1 = z2;
theorem :: SEQ_4:65
z1 + z = z2 + z or z1 + z = z + z2 implies z1 = z2;
theorem :: SEQ_4:66
-(z1 + z2) = -z1 + -z2;
definition let n,z1,z2;
redefine func z1 - z2 -> Element of COMPLEX n;
end;
theorem :: SEQ_4:67
k in Seg n implies (z1 - z2).k = z1.k - z2.k;
theorem :: SEQ_4:68
z - 0c n = z;
theorem :: SEQ_4:69
0c n - z = -z;
theorem :: SEQ_4:70
z1 - -z2 = z1 + z2;
theorem :: SEQ_4:71
-(z1 - z2) = z2 - z1;
theorem :: SEQ_4:72
-(z1 - z2) = -z1 + z2;
theorem :: SEQ_4:73
z - z = 0c n;
theorem :: SEQ_4:74
z1 - z2 = 0c n implies z1 = z2;
theorem :: SEQ_4:75
z1 - z2 - z3 = z1 - (z2 + z3);
theorem :: SEQ_4:76
z1 + (z2 - z3) = z1 + z2 - z3;
theorem :: SEQ_4:77
z1 - (z2 - z3) = z1 - z2 + z3;
theorem :: SEQ_4:78
z1 - z2 + z3 = z1 + z3 - z2;
theorem :: SEQ_4:79
z1 = z1 + z - z;
theorem :: SEQ_4:80
z1 + (z2 - z1) = z2;
theorem :: SEQ_4:81
z1 = z1 - z + z;
definition let n,z,c;
redefine func c*z -> Element of COMPLEX n;
end;
theorem :: SEQ_4:82
k in Seg n & c9 = z.k implies (c*z).k = c*c9;
theorem :: SEQ_4:83
c1*(c2*z) = (c1*c2)*z;
theorem :: SEQ_4:84
(c1 + c2)*z = c1*z + c2*z;
theorem :: SEQ_4:85
c*(z1+z2) = c*z1 + c*z2;
theorem :: SEQ_4:86
1r*z = z;
theorem :: SEQ_4:87
0c*z = 0c n;
theorem :: SEQ_4:88
(-1r)*z = -z;
definition let n,z;
redefine func abs z -> Element of n-tuples_on REAL;
end;
theorem :: SEQ_4:89
k in Seg n & c = z.k implies (abs z).k = |.c.|;
theorem :: SEQ_4:90
abs 0c n = n |-> 0;
theorem :: SEQ_4:91
abs -z = abs z;
theorem :: SEQ_4:92
abs(c*z) = |.c.|*(abs z);
definition
let z be FinSequence of COMPLEX;
func |.z.| -> Real equals
:: SEQ_4:def 13
sqrt Sum sqr abs z;
end;
theorem :: SEQ_4:93
|.0c n.| = 0;
theorem :: SEQ_4:94
|.z.| = 0 implies z = 0c n;
theorem :: SEQ_4:95
0 <= |.z.|;
theorem :: SEQ_4:96
|.-z.| = |.z.|;
theorem :: SEQ_4:97
|.c*z.| = |.c.|*|.z.|;
theorem :: SEQ_4:98
|.z1 + z2.| <= |.z1.| + |.z2.|;
theorem :: SEQ_4:99
|.z1 - z2.| <= |.z1.| + |.z2.|;
theorem :: SEQ_4:100
|.z1.| - |.z2.| <= |.z1 + z2.|;
theorem :: SEQ_4:101
|.z1.| - |.z2.| <= |.z1 - z2.|;
theorem :: SEQ_4:102
|.z1 - z2.| = 0 iff z1 = z2;
theorem :: SEQ_4:103
z1 <> z2 implies 0 < |.z1 - z2.|;
theorem :: SEQ_4:104
|.z1 - z2.| = |.z2 - z1.|;
theorem :: SEQ_4:105
|.z1 - z2.| <= |.z1 - z.| + |.z - z2.|;
definition
let n;
let A be Subset of COMPLEX n;
attr A is open means
:: SEQ_4:def 14
for x st x in A ex r st 0 < r & for z st |.z.| < r holds x + z in A;
end;
definition let n;
let A be Subset of COMPLEX n;
attr A is closed means
:: SEQ_4:def 15
for x st for r st r > 0 ex z st |.z.| < r & x + z in A holds x in A;
end;
theorem :: SEQ_4:106
for A being Subset of COMPLEX n st A = {} holds A is open;
theorem :: SEQ_4:107
for A being Subset of COMPLEX n st A = COMPLEX n holds A is open;
theorem :: SEQ_4:108
for AA being Subset-Family of COMPLEX n st for A being Subset of
COMPLEX n st A in AA holds A is open for A being Subset of COMPLEX n st A =
union AA holds A is open;
theorem :: SEQ_4:109
for A,B being Subset of COMPLEX n st A is open & B is open for C
being Subset of COMPLEX n st C = A /\ B holds C is open;
definition
let n,x,r;
func Ball(x,r) -> Subset of COMPLEX n equals
:: SEQ_4:def 16
{ z : |.z - x.| < r };
end;
theorem :: SEQ_4:110
z in Ball(x,r) iff |.x - z.| < r;
theorem :: SEQ_4:111
0 < r implies x in Ball(x,r);
theorem :: SEQ_4:112
Ball(z1,r1) is open;
definition let n,x,A;
func dist(x,A) -> Real means
:: SEQ_4:def 17
for X being Subset of REAL st X = {|.x
- z.| : z in A} holds it = lower_bound X;
end;
definition let n,A,r;
func Ball(A,r) -> Subset of COMPLEX n equals
:: SEQ_4:def 18
{ z : dist(z,A) < r };
end;
theorem :: SEQ_4:113
for X being Subset of REAL, r st X <> {} & for r9 st r9 in X
holds r <= r9 holds lower_bound X >= r;
theorem :: SEQ_4:114
A <> {} implies dist(x,A) >= 0;
theorem :: SEQ_4:115
A <> {} implies dist(x + z,A) <= dist(x,A) + |.z.|;
theorem :: SEQ_4:116
x in A implies dist(x,A) = 0;
theorem :: SEQ_4:117
not x in A & A <> {} & A is closed implies dist(x,A) > 0;
theorem :: SEQ_4:118
A <> {} implies |.z1 - x.| + dist(x,A) >= dist(z1,A);
theorem :: SEQ_4:119
z in Ball(A,r) iff dist(z,A) < r;
theorem :: SEQ_4:120
0 < r & x in A implies x in Ball(A,r);
theorem :: SEQ_4:121
0 < r implies A c= Ball(A,r);
theorem :: SEQ_4:122
A <> {} implies Ball(A,r1) is open;
definition
let n,A,B;
func dist(A,B) -> Real means
:: SEQ_4:def 19
for X being Subset of REAL st X = {|.x
- z.| : x in A & z in B} holds it = lower_bound X;
end;
theorem :: SEQ_4:123
for X,Y being Subset of REAL holds X <> {} & Y <> {} implies X ++ Y <> {};
theorem :: SEQ_4:124
for X,Y being Subset of REAL holds X is bounded_below &
Y is bounded_below implies X++Y is bounded_below;
theorem :: SEQ_4:125
for X,Y being Subset of REAL st
X <> {} & X is bounded_below & Y <> {} & Y is bounded_below holds
lower_bound (X ++ Y) = lower_bound X + lower_bound Y;
theorem :: SEQ_4:126
for X,Y being Subset of REAL st Y is bounded_below & X <> {} &
for r st r in X ex r1 st r1 in Y & r1 <= r holds
lower_bound X >= lower_bound Y;
theorem :: SEQ_4:127
A <> {} & B <> {} implies dist(A,B) >= 0;
theorem :: SEQ_4:128
dist(A,B) = dist(B,A);
theorem :: SEQ_4:129
A <> {} & B <> {} implies dist(x,A) + dist(x,B) >= dist(A,B);
theorem :: SEQ_4:130
A meets B implies dist(A,B) = 0;
definition
let n;
func ComplexOpenSets(n) -> Subset-Family of COMPLEX n equals
:: SEQ_4:def 20
{A where A is
Subset of COMPLEX n: A is open};
end;
theorem :: SEQ_4:131
for A being Subset of COMPLEX n holds A in ComplexOpenSets n iff A is open;
theorem :: SEQ_4:132
for A being Subset of COMPLEX n holds A is closed iff A` is open;
begin :: moved from GOBOARD2, 2010.03.01, A.T.
reserve
v,v1,v2 for FinSequence of REAL,
n,m,k for Nat,
x for set;
theorem :: SEQ_4:133
for R being finite Subset of REAL holds R <> {} implies R is
bounded_above & upper_bound(R) in R & R is bounded_below & lower_bound(R) in R;
theorem :: SEQ_4:134
for n being Nat for f being FinSequence holds 1 <= n & n+1 <= len
f iff n in dom f & n+1 in dom f;
theorem :: SEQ_4:135
for n being Nat for f being FinSequence holds 1 <= n & n+2 <= len f
iff n in dom f & n+1 in dom f & n+2 in dom f;
theorem :: SEQ_4:136
for D being non empty set, f1,f2 being FinSequence of D, n st 1
<= n & n <= len f2 holds (f1^f2)/.(n + len f1) = f2/.n;
theorem :: SEQ_4:137
v is increasing implies for n,m st n in dom v & m in dom v & n<=m
holds v.n <= v.m;
theorem :: SEQ_4:138
v is increasing implies for n,m st n in dom v & m in dom v & n<>
m holds v.n<>v.m;
theorem :: SEQ_4:139
v is increasing & v1 = v|Seg n implies v1 is increasing;
theorem :: SEQ_4:140
for v holds ex v1 st rng v1 = rng v & len v1 = card rng v & v1 is
increasing;
theorem :: SEQ_4:141
for v1,v2 st len v1 = len v2 & rng v1 = rng v2 & v1 is increasing & v2
is increasing holds v1 = v2;
definition
let v;
func Incr(v) ->increasing FinSequence of REAL means
:: SEQ_4:def 21
rng it = rng v & len it = card rng v;
end;
registration
let v be non empty FinSequence of REAL;
cluster Incr v -> non empty;
end;
:: from SPRECT_1, 2011.04.24, A.T
registration
cluster non empty bounded_above bounded_below for Subset of REAL;
end;
theorem :: SEQ_4:142
for A,B being non empty bounded_below Subset of REAL holds lower_bound(A
\/ B) = min(lower_bound A,lower_bound B);
theorem :: SEQ_4:143
for A,B being non empty bounded_above Subset of REAL holds upper_bound(A
\/ B) = max(upper_bound A,upper_bound B);
:: from TOPMETR3, 2011.04.24, A.T
theorem :: SEQ_4:144
for R being non empty Subset of REAL,r0 being Real st for
r being Real st r in R holds r <= r0 holds upper_bound R <= r0;
~~