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;