Article RINFSUP2, MML version 4.99.1005
:: RINFSUP2:th 1
theorem
for b1 being non empty Element of bool ExtREAL
for b2 being non empty Element of bool REAL
st b1 = b2 & b2 is bounded_above
holds b1 is bounded_above & sup b1 = sup b2;
:: RINFSUP2:th 2
theorem
for b1 being non empty Element of bool ExtREAL
for b2 being non empty Element of bool REAL
st b1 = b2 & b1 is bounded_above
holds b2 is bounded_above & sup b1 = sup b2;
:: RINFSUP2:th 3
theorem
for b1 being non empty Element of bool ExtREAL
for b2 being non empty Element of bool REAL
st b1 = b2 & b2 is bounded_below
holds b1 is bounded_below & inf b1 = inf b2;
:: RINFSUP2:th 4
theorem
for b1 being non empty Element of bool ExtREAL
for b2 being non empty Element of bool REAL
st b1 = b2 & b1 is bounded_below
holds b2 is bounded_below & inf b1 = inf b2;
:: RINFSUP2:funcnot 1 => RINFSUP2:func 1
definition
let a1 be Function-like quasi_total Relation of NAT,ExtREAL;
func sup A1 -> Element of ExtREAL equals
sup rng a1;
end;
:: RINFSUP2:def 1
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL holds
sup b1 = sup rng b1;
:: RINFSUP2:funcnot 2 => RINFSUP2:func 2
definition
let a1 be Function-like quasi_total Relation of NAT,ExtREAL;
func inf A1 -> Element of ExtREAL equals
inf rng a1;
end;
:: RINFSUP2:def 2
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL holds
inf b1 = inf rng b1;
:: RINFSUP2:attrnot 1 => RINFSUP2:attr 1
definition
let a1 be Function-like quasi_total Relation of NAT,ExtREAL;
attr a1 is bounded_below means
rng a1 is bounded_below;
end;
:: RINFSUP2:dfs 3
definiens
let a1 be Function-like quasi_total Relation of NAT,ExtREAL;
To prove
a1 is bounded_below
it is sufficient to prove
thus rng a1 is bounded_below;
:: RINFSUP2:def 3
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL holds
b1 is bounded_below
iff
rng b1 is bounded_below;
:: RINFSUP2:attrnot 2 => RINFSUP2:attr 2
definition
let a1 be Function-like quasi_total Relation of NAT,ExtREAL;
attr a1 is bounded_above means
rng a1 is bounded_above;
end;
:: RINFSUP2:dfs 4
definiens
let a1 be Function-like quasi_total Relation of NAT,ExtREAL;
To prove
a1 is bounded_above
it is sufficient to prove
thus rng a1 is bounded_above;
:: RINFSUP2:def 4
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL holds
b1 is bounded_above
iff
rng b1 is bounded_above;
:: RINFSUP2:attrnot 3 => RINFSUP2:attr 3
definition
let a1 be Function-like quasi_total Relation of NAT,ExtREAL;
attr a1 is bounded means
a1 is bounded_above & a1 is bounded_below;
end;
:: RINFSUP2:dfs 5
definiens
let a1 be Function-like quasi_total Relation of NAT,ExtREAL;
To prove
a1 is bounded
it is sufficient to prove
thus a1 is bounded_above & a1 is bounded_below;
:: RINFSUP2:def 5
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL holds
b1 is bounded
iff
b1 is bounded_above & b1 is bounded_below;
:: RINFSUP2:th 5
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL
for b2 being Element of NAT holds
{b1 . b3 where b3 is Element of NAT: b2 <= b3} is non empty Element of bool ExtREAL;
:: RINFSUP2:funcnot 3 => RINFSUP2:func 3
definition
let a1 be Function-like quasi_total Relation of NAT,ExtREAL;
func inferior_realsequence A1 -> Function-like quasi_total Relation of NAT,ExtREAL means
for b1 being Element of NAT holds
ex b2 being non empty Element of bool ExtREAL st
b2 = {a1 . b3 where b3 is Element of NAT: b1 <= b3} &
it . b1 = inf b2;
end;
:: RINFSUP2:def 6
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,ExtREAL holds
b2 = inferior_realsequence b1
iff
for b3 being Element of NAT holds
ex b4 being non empty Element of bool ExtREAL st
b4 = {b1 . b5 where b5 is Element of NAT: b3 <= b5} &
b2 . b3 = inf b4;
:: RINFSUP2:funcnot 4 => RINFSUP2:func 4
definition
let a1 be Function-like quasi_total Relation of NAT,ExtREAL;
func superior_realsequence A1 -> Function-like quasi_total Relation of NAT,ExtREAL means
for b1 being Element of NAT holds
ex b2 being non empty Element of bool ExtREAL st
b2 = {a1 . b3 where b3 is Element of NAT: b1 <= b3} &
it . b1 = sup b2;
end;
:: RINFSUP2:def 7
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,ExtREAL holds
b2 = superior_realsequence b1
iff
for b3 being Element of NAT holds
ex b4 being non empty Element of bool ExtREAL st
b4 = {b1 . b5 where b5 is Element of NAT: b3 <= b5} &
b2 . b3 = sup b4;
:: RINFSUP2:th 6
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL
st b1 is real-valued
holds b1 is Function-like quasi_total Relation of NAT,REAL;
:: RINFSUP2:attrnot 4 => RINFSUP2:attr 4
definition
let a1 be Function-like Relation of NAT,ExtREAL;
attr a1 is increasing means
for b1, b2 being Element of NAT
st b1 in dom a1 & b2 in dom a1 & b1 < b2
holds a1 . b1 < a1 . b2;
end;
:: RINFSUP2:dfs 8
definiens
let a1 be Function-like Relation of NAT,ExtREAL;
To prove
a1 is increasing
it is sufficient to prove
thus for b1, b2 being Element of NAT
st b1 in dom a1 & b2 in dom a1 & b1 < b2
holds a1 . b1 < a1 . b2;
:: RINFSUP2:def 8
theorem
for b1 being Function-like Relation of NAT,ExtREAL holds
b1 is increasing
iff
for b2, b3 being Element of NAT
st b2 in dom b1 & b3 in dom b1 & b2 < b3
holds b1 . b2 < b1 . b3;
:: RINFSUP2:attrnot 5 => RINFSUP2:attr 5
definition
let a1 be Function-like Relation of NAT,ExtREAL;
attr a1 is decreasing means
for b1, b2 being Element of NAT
st b1 in dom a1 & b2 in dom a1 & b1 < b2
holds a1 . b2 < a1 . b1;
end;
:: RINFSUP2:dfs 9
definiens
let a1 be Function-like Relation of NAT,ExtREAL;
To prove
a1 is decreasing
it is sufficient to prove
thus for b1, b2 being Element of NAT
st b1 in dom a1 & b2 in dom a1 & b1 < b2
holds a1 . b2 < a1 . b1;
:: RINFSUP2:def 9
theorem
for b1 being Function-like Relation of NAT,ExtREAL holds
b1 is decreasing
iff
for b2, b3 being Element of NAT
st b2 in dom b1 & b3 in dom b1 & b2 < b3
holds b1 . b3 < b1 . b2;
:: RINFSUP2:attrnot 6 => RINFSUP2:attr 6
definition
let a1 be Function-like Relation of NAT,ExtREAL;
attr a1 is non-decreasing means
for b1, b2 being Element of NAT
st b1 in dom a1 & b2 in dom a1 & b1 <= b2
holds a1 . b1 <= a1 . b2;
end;
:: RINFSUP2:dfs 10
definiens
let a1 be Function-like Relation of NAT,ExtREAL;
To prove
a1 is non-decreasing
it is sufficient to prove
thus for b1, b2 being Element of NAT
st b1 in dom a1 & b2 in dom a1 & b1 <= b2
holds a1 . b1 <= a1 . b2;
:: RINFSUP2:def 10
theorem
for b1 being Function-like Relation of NAT,ExtREAL holds
b1 is non-decreasing
iff
for b2, b3 being Element of NAT
st b2 in dom b1 & b3 in dom b1 & b2 <= b3
holds b1 . b2 <= b1 . b3;
:: RINFSUP2:attrnot 7 => RINFSUP2:attr 7
definition
let a1 be Function-like Relation of NAT,ExtREAL;
attr a1 is non-increasing means
for b1, b2 being Element of NAT
st b1 in dom a1 & b2 in dom a1 & b1 <= b2
holds a1 . b2 <= a1 . b1;
end;
:: RINFSUP2:dfs 11
definiens
let a1 be Function-like Relation of NAT,ExtREAL;
To prove
a1 is non-increasing
it is sufficient to prove
thus for b1, b2 being Element of NAT
st b1 in dom a1 & b2 in dom a1 & b1 <= b2
holds a1 . b2 <= a1 . b1;
:: RINFSUP2:def 11
theorem
for b1 being Function-like Relation of NAT,ExtREAL holds
b1 is non-increasing
iff
for b2, b3 being Element of NAT
st b2 in dom b1 & b3 in dom b1 & b2 <= b3
holds b1 . b3 <= b1 . b2;
:: RINFSUP2:th 7
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL holds
(b1 is increasing implies for b2, b3 being Element of NAT
st b3 < b2
holds b1 . b3 < b1 . b2) &
(for b2, b3 being Element of NAT
st b3 < b2
holds b1 . b3 < b1 . b2 implies b1 is increasing) &
(b1 is decreasing implies for b2, b3 being Element of NAT
st b3 < b2
holds b1 . b2 < b1 . b3) &
(for b2, b3 being Element of NAT
st b3 < b2
holds b1 . b2 < b1 . b3 implies b1 is decreasing) &
(b1 is non-decreasing implies for b2, b3 being Element of NAT
st b3 <= b2
holds b1 . b3 <= b1 . b2) &
(for b2, b3 being Element of NAT
st b3 <= b2
holds b1 . b3 <= b1 . b2 implies b1 is non-decreasing) &
(b1 is non-increasing implies for b2, b3 being Element of NAT
st b3 <= b2
holds b1 . b2 <= b1 . b3) &
(for b2, b3 being Element of NAT
st b3 <= b2
holds b1 . b2 <= b1 . b3 implies b1 is non-increasing);
:: RINFSUP2:th 8
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,ExtREAL holds
(inferior_realsequence b2) . b1 <= b2 . b1 & b2 . b1 <= (superior_realsequence b2) . b1;
:: RINFSUP2:funcreg 1
registration
let a1 be Function-like quasi_total Relation of NAT,ExtREAL;
cluster superior_realsequence a1 -> Function-like quasi_total non-increasing;
end;
:: RINFSUP2:funcreg 2
registration
let a1 be Function-like quasi_total Relation of NAT,ExtREAL;
cluster inferior_realsequence a1 -> Function-like quasi_total non-decreasing;
end;
:: RINFSUP2:funcnot 5 => RINFSUP2:func 5
definition
let a1 be Function-like quasi_total Relation of NAT,ExtREAL;
func lim_sup A1 -> Element of ExtREAL equals
inf superior_realsequence a1;
end;
:: RINFSUP2:def 12
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL holds
lim_sup b1 = inf superior_realsequence b1;
:: RINFSUP2:funcnot 6 => RINFSUP2:func 6
definition
let a1 be Function-like quasi_total Relation of NAT,ExtREAL;
func lim_inf A1 -> Element of ExtREAL equals
sup inferior_realsequence a1;
end;
:: RINFSUP2:def 13
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL holds
lim_inf b1 = sup inferior_realsequence b1;
:: RINFSUP2:th 9
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL
for b2 being Function-like quasi_total Relation of NAT,REAL
st b1 = b2 & b2 is bounded
holds superior_realsequence b1 = superior_realsequence b2 & lim_sup b1 = lim_sup b2;
:: RINFSUP2:th 10
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL
for b2 being Function-like quasi_total Relation of NAT,REAL
st b1 = b2 & b2 is bounded
holds inferior_realsequence b1 = inferior_realsequence b2 & lim_inf b1 = lim_inf b2;
:: RINFSUP2:th 11
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL
st b1 is bounded
holds b1 is Function-like quasi_total Relation of NAT,REAL;
:: RINFSUP2:th 12
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL
for b2 being Function-like quasi_total Relation of NAT,REAL
st b1 = b2
holds b1 is bounded_above
iff
b2 is bounded_above;
:: RINFSUP2:th 13
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL
for b2 being Function-like quasi_total Relation of NAT,REAL
st b1 = b2
holds b1 is bounded_below
iff
b2 is bounded_below;
:: RINFSUP2:th 14
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL
for b2 being Function-like quasi_total Relation of NAT,REAL
st b1 = b2 & b2 is convergent
holds b1 is convergent_to_finite_number & b1 is convergent & lim b1 = lim b2;
:: RINFSUP2:th 15
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL
for b2 being Function-like quasi_total Relation of NAT,REAL
st b1 = b2 & b1 is convergent_to_finite_number
holds b2 is convergent & lim b1 = lim b2;
:: RINFSUP2:th 16
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,ExtREAL
st b2 ^\ b1 is convergent_to_finite_number
holds b2 is convergent_to_finite_number & b2 is convergent & lim b2 = lim (b2 ^\ b1);
:: RINFSUP2:th 17
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,ExtREAL
st b2 ^\ b1 is convergent
holds b2 is convergent & lim b2 = lim (b2 ^\ b1);
:: RINFSUP2:th 18
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL
st lim_sup b1 = lim_inf b1 & lim_inf b1 in REAL
holds ex b2 being Element of NAT st
b1 ^\ b2 is bounded;
:: RINFSUP2:th 19
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL
st b1 is convergent_to_finite_number
holds ex b2 being Element of NAT st
b1 ^\ b2 is bounded;
:: RINFSUP2:th 20
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,ExtREAL
st b2 is convergent_to_finite_number
holds b2 ^\ b1 is convergent_to_finite_number & b2 ^\ b1 is convergent & lim b2 = lim (b2 ^\ b1);
:: RINFSUP2:th 21
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,ExtREAL
st b2 is convergent
holds b2 ^\ b1 is convergent & lim b2 = lim (b2 ^\ b1);
:: RINFSUP2:th 22
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,ExtREAL holds
(b2 is bounded_above implies b2 ^\ b1 is bounded_above) & (b2 is bounded_below implies b2 ^\ b1 is bounded_below);
:: RINFSUP2:th 23
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,ExtREAL holds
inf b2 <= b2 . b1 & b2 . b1 <= sup b2;
:: RINFSUP2:th 24
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL holds
inf b1 <= sup b1;
:: RINFSUP2:th 25
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,ExtREAL
st b2 is non-increasing
holds b2 ^\ b1 is non-increasing & inf b2 = inf (b2 ^\ b1);
:: RINFSUP2:th 26
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,ExtREAL
st b2 is non-decreasing
holds b2 ^\ b1 is non-decreasing & sup b2 = sup (b2 ^\ b1);
:: RINFSUP2:th 27
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,ExtREAL holds
(superior_realsequence b2) . b1 = sup (b2 ^\ b1) &
(inferior_realsequence b2) . b1 = inf (b2 ^\ b1);
:: RINFSUP2:th 28
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL
for b2 being Element of NAT holds
superior_realsequence (b1 ^\ b2) = (superior_realsequence b1) ^\ b2 &
lim_sup (b1 ^\ b2) = lim_sup b1;
:: RINFSUP2:th 29
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL
for b2 being Element of NAT holds
inferior_realsequence (b1 ^\ b2) = (inferior_realsequence b1) ^\ b2 &
lim_inf (b1 ^\ b2) = lim_inf b1;
:: RINFSUP2:th 30
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL
for b2 being Element of NAT
st b1 is non-increasing & -infty < b1 . b2 & b1 . b2 < +infty
holds b1 ^\ b2 is bounded_above & sup (b1 ^\ b2) = b1 . b2;
:: RINFSUP2:th 31
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL
for b2 being Element of NAT
st b1 is non-decreasing & -infty < b1 . b2 & b1 . b2 < +infty
holds b1 ^\ b2 is bounded_below & inf (b1 ^\ b2) = b1 . b2;
:: RINFSUP2:th 32
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL
st for b2 being Element of NAT holds
+infty <= b1 . b2
holds b1 is convergent_to_+infty;
:: RINFSUP2:th 33
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL
st for b2 being Element of NAT holds
b1 . b2 <= -infty
holds b1 is convergent_to_-infty;
:: RINFSUP2:th 34
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL
st b1 is non-increasing & -infty = inf b1
holds b1 is convergent_to_-infty & lim b1 = -infty;
:: RINFSUP2:th 35
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL
st b1 is non-decreasing & +infty = sup b1
holds b1 is convergent_to_+infty & lim b1 = +infty;
:: RINFSUP2:th 36
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL
st b1 is non-increasing
holds b1 is convergent & lim b1 = inf b1;
:: RINFSUP2:th 37
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL
st b1 is non-decreasing
holds b1 is convergent & lim b1 = sup b1;
:: RINFSUP2:th 38
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,ExtREAL
st b1 is convergent &
b2 is convergent &
(for b3 being Element of NAT holds
b1 . b3 <= b2 . b3)
holds lim b1 <= lim b2;
:: RINFSUP2:th 39
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL holds
lim_inf b1 <= lim_sup b1;
:: RINFSUP2:th 40
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL holds
b1 is convergent
iff
lim_inf b1 = lim_sup b1;
:: RINFSUP2:th 41
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL
st b1 is convergent
holds lim b1 = lim_inf b1 & lim b1 = lim_sup b1;