Article SEQ_2, MML version 4.99.1005
:: SEQ_2:th 3
theorem
for b1 being real set
st 0 < b1
holds 0 < b1 / 2 & 0 < b1 / 4;
:: SEQ_2:th 9
theorem
for b1, b2 being real set holds
- b1 < b2 & b2 < b1
iff
abs b2 < b1;
:: SEQ_2:th 11
theorem
for b1, b2 being real set
st b1 <> 0 & b2 <> 0
holds abs (b1 " - (b2 ")) = (abs (b1 - b2)) / ((abs b1) * abs b2);
:: SEQ_2:attrnot 1 => SEQ_2:attr 1
definition
let a1 be Relation-like Function-like real-valued set;
attr a1 is bounded_above means
ex b1 being real set st
for b2 being set
st b2 in proj1 a1
holds a1 . b2 < b1;
end;
:: SEQ_2:dfs 1
definiens
let a1 be Relation-like Function-like real-valued set;
To prove
a1 is bounded_above
it is sufficient to prove
thus ex b1 being real set st
for b2 being set
st b2 in proj1 a1
holds a1 . b2 < b1;
:: SEQ_2:def 1
theorem
for b1 being Relation-like Function-like real-valued set holds
b1 is bounded_above
iff
ex b2 being real set st
for b3 being set
st b3 in proj1 b1
holds b1 . b3 < b2;
:: SEQ_2:attrnot 2 => SEQ_2:attr 2
definition
let a1 be Relation-like Function-like real-valued set;
attr a1 is bounded_below means
ex b1 being real set st
for b2 being set
st b2 in proj1 a1
holds b1 < a1 . b2;
end;
:: SEQ_2:dfs 2
definiens
let a1 be Relation-like Function-like real-valued set;
To prove
a1 is bounded_below
it is sufficient to prove
thus ex b1 being real set st
for b2 being set
st b2 in proj1 a1
holds b1 < a1 . b2;
:: SEQ_2:def 2
theorem
for b1 being Relation-like Function-like real-valued set holds
b1 is bounded_below
iff
ex b2 being real set st
for b3 being set
st b3 in proj1 b1
holds b2 < b1 . b3;
:: SEQ_2:attrnot 3 => SEQ_2:attr 1
definition
let a1 be Relation-like Function-like real-valued set;
attr a1 is bounded_above means
ex b1 being real set st
for b2 being Element of NAT holds
a1 . b2 < b1;
end;
:: SEQ_2:dfs 3
definiens
let a1 be Function-like quasi_total Relation of NAT,REAL;
To prove
a1 is bounded_above
it is sufficient to prove
thus ex b1 being real set st
for b2 being Element of NAT holds
a1 . b2 < b1;
:: SEQ_2:def 3
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
b1 is bounded_above
iff
ex b2 being real set st
for b3 being Element of NAT holds
b1 . b3 < b2;
:: SEQ_2:attrnot 4 => SEQ_2:attr 2
definition
let a1 be Relation-like Function-like real-valued set;
attr a1 is bounded_below means
ex b1 being real set st
for b2 being Element of NAT holds
b1 < a1 . b2;
end;
:: SEQ_2:dfs 4
definiens
let a1 be Function-like quasi_total Relation of NAT,REAL;
To prove
a1 is bounded_below
it is sufficient to prove
thus ex b1 being real set st
for b2 being Element of NAT holds
b1 < a1 . b2;
:: SEQ_2:def 4
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
b1 is bounded_below
iff
ex b2 being real set st
for b3 being Element of NAT holds
b2 < b1 . b3;
:: SEQ_2:attrnot 5 => SEQ_2:attr 3
definition
let a1 be Relation-like Function-like real-valued set;
attr a1 is bounded means
a1 is bounded_above & a1 is bounded_below;
end;
:: SEQ_2:dfs 5
definiens
let a1 be Relation-like Function-like real-valued set;
To prove
a1 is bounded
it is sufficient to prove
thus a1 is bounded_above & a1 is bounded_below;
:: SEQ_2:def 5
theorem
for b1 being Relation-like Function-like real-valued set holds
b1 is bounded
iff
b1 is bounded_above & b1 is bounded_below;
:: SEQ_2:condreg 1
registration
cluster Relation-like Function-like real-valued bounded -> bounded_above bounded_below (set);
end;
:: SEQ_2:condreg 2
registration
cluster Relation-like Function-like real-valued bounded_above bounded_below -> bounded (set);
end;
:: SEQ_2:th 15
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
b1 is bounded
iff
ex b2 being real set st
0 < b2 &
(for b3 being Element of NAT holds
abs (b1 . b3) < b2);
:: SEQ_2:th 16
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being Element of NAT holds
ex b3 being real set st
0 < b3 &
(for b4 being Element of NAT
st b4 <= b2
holds abs (b1 . b4) < b3);
:: SEQ_2:attrnot 6 => SEQ_2:attr 4
definition
let a1 be Function-like quasi_total Relation of NAT,REAL;
attr a1 is convergent means
ex b1 being real set st
for b2 being real set
st 0 < b2
holds ex b3 being Element of NAT st
for b4 being Element of NAT
st b3 <= b4
holds abs ((a1 . b4) - b1) < b2;
end;
:: SEQ_2:dfs 6
definiens
let a1 be Function-like quasi_total Relation of NAT,REAL;
To prove
a1 is convergent
it is sufficient to prove
thus ex b1 being real set st
for b2 being real set
st 0 < b2
holds ex b3 being Element of NAT st
for b4 being Element of NAT
st b3 <= b4
holds abs ((a1 . b4) - b1) < b2;
:: SEQ_2:def 6
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
b1 is convergent
iff
ex b2 being real set st
for b3 being real set
st 0 < b3
holds ex b4 being Element of NAT st
for b5 being Element of NAT
st b4 <= b5
holds abs ((b1 . b5) - b2) < b3;
:: SEQ_2:funcnot 1 => SEQ_2:func 1
definition
let a1 be Function-like quasi_total Relation of NAT,REAL;
assume a1 is convergent;
func lim A1 -> real set means
for b1 being real set
st 0 < b1
holds ex b2 being Element of NAT st
for b3 being Element of NAT
st b2 <= b3
holds abs ((a1 . b3) - it) < b1;
end;
:: SEQ_2:def 7
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
st b1 is convergent
for b2 being real set holds
b2 = lim b1
iff
for b3 being real set
st 0 < b3
holds ex b4 being Element of NAT st
for b5 being Element of NAT
st b4 <= b5
holds abs ((b1 . b5) - b2) < b3;
:: SEQ_2:funcnot 2 => SEQ_2:func 2
definition
let a1 be Function-like quasi_total Relation of NAT,REAL;
redefine func lim a1 -> Element of REAL;
end;
:: SEQ_2:th 19
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
st b1 is convergent & b2 is convergent
holds b1 + b2 is convergent;
:: SEQ_2:th 20
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
st b1 is convergent & b2 is convergent
holds lim (b1 + b2) = (lim b1) + lim b2;
:: SEQ_2:th 21
theorem
for b1 being real set
for b2 being Function-like quasi_total Relation of NAT,REAL
st b2 is convergent
holds b1 (#) b2 is convergent;
:: SEQ_2:th 22
theorem
for b1 being real set
for b2 being Function-like quasi_total Relation of NAT,REAL
st b2 is convergent
holds lim (b1 (#) b2) = b1 * lim b2;
:: SEQ_2:th 23
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
st b1 is convergent
holds - b1 is convergent;
:: SEQ_2:th 24
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
st b1 is convergent
holds lim - b1 = - lim b1;
:: SEQ_2:th 25
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
st b1 is convergent & b2 is convergent
holds b1 - b2 is convergent;
:: SEQ_2:th 26
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
st b1 is convergent & b2 is convergent
holds lim (b1 - b2) = (lim b1) - lim b2;
:: SEQ_2:th 27
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
st b1 is convergent
holds b1 is bounded;
:: SEQ_2:th 28
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
st b1 is convergent & b2 is convergent
holds b1 (#) b2 is convergent;
:: SEQ_2:th 29
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
st b1 is convergent & b2 is convergent
holds lim (b1 (#) b2) = (lim b1) * lim b2;
:: SEQ_2:th 30
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
st b1 is convergent & lim b1 <> 0
holds ex b2 being Element of NAT st
for b3 being Element of NAT
st b2 <= b3
holds (abs lim b1) / 2 < abs (b1 . b3);
:: SEQ_2:th 31
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
st b1 is convergent &
(for b2 being Element of NAT holds
0 <= b1 . b2)
holds 0 <= lim b1;
:: SEQ_2:th 32
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
st b1 is convergent &
b2 is convergent &
(for b3 being Element of NAT holds
b1 . b3 <= b2 . b3)
holds lim b1 <= lim b2;
:: SEQ_2:th 33
theorem
for b1, b2, b3 being Function-like quasi_total Relation of NAT,REAL
st b1 is convergent &
b2 is convergent &
(for b4 being Element of NAT holds
b1 . b4 <= b3 . b4 & b3 . b4 <= b2 . b4) &
lim b1 = lim b2
holds b3 is convergent;
:: SEQ_2:th 34
theorem
for b1, b2, b3 being Function-like quasi_total Relation of NAT,REAL
st b1 is convergent &
b2 is convergent &
(for b4 being Element of NAT holds
b1 . b4 <= b3 . b4 & b3 . b4 <= b2 . b4) &
lim b1 = lim b2
holds lim b3 = lim b1;
:: SEQ_2:th 35
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
st b1 is convergent & lim b1 <> 0 & b1 is non-empty
holds b1 " is convergent;
:: SEQ_2:th 36
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
st b1 is convergent & lim b1 <> 0 & b1 is non-empty
holds lim (b1 ") = (lim b1) ";
:: SEQ_2:th 37
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
st b1 is convergent & b2 is convergent & lim b2 <> 0 & b2 is non-empty
holds b1 /" b2 is convergent;
:: SEQ_2:th 38
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
st b1 is convergent & b2 is convergent & lim b2 <> 0 & b2 is non-empty
holds lim (b1 /" b2) = (lim b1) / lim b2;
:: SEQ_2:th 39
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
st b1 is convergent & b2 is bounded & lim b1 = 0
holds b1 (#) b2 is convergent;
:: SEQ_2:th 40
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
st b1 is convergent & b2 is bounded & lim b1 = 0
holds lim (b1 (#) b2) = 0;