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;