reserve X for non empty set,
  S for SigmaField of X,
  M for sigma_Measure of S,
  E for Element of S,
  F,G for Functional_Sequence of X,ExtREAL,
  I for ExtREAL_sequence,
  f,g for PartFunc of X,ExtREAL,
  seq, seq1, seq2 for ExtREAL_sequence,
  p for ExtReal,
  n,m for Nat,
  x for Element of X,
  z,D for set;

theorem Th7:
  ( seq is convergent_to_finite_number implies ex g be Real
st lim seq = g & for p be Real st 0<p ex n be Nat st for m be Nat st n<=
m holds |. seq.m - lim seq .| < p ) & ( seq is convergent_to_+infty implies lim
  seq = +infty ) & ( seq is convergent_to_-infty implies lim seq = -infty )
proof
A1: now
    assume
A2: seq is convergent_to_finite_number;
    then
A3: not seq is convergent_to_+infty by MESFUNC5:50;
A4: not seq is convergent_to_-infty by A2,MESFUNC5:51;
    seq is convergent by A2;
    then consider g be Real such that
A5: lim seq = g and
A6: for p be Real st 0<p ex n be Nat st for m be Nat st n<=m
    holds |.seq.m - lim seq.| < p and
    seq is convergent_to_finite_number by A3,A4,MESFUNC5:def 12;
    take g;
    thus ex g be Real st lim seq = g & for p be Real st 0<p ex n
    be Nat st for m be Nat st n<=m holds |. seq.m - lim seq .| < p by A5,A6;
  end;
A7: now
    assume
A8: seq is convergent_to_-infty;
    then seq is convergent;
    hence lim seq = -infty by A8,MESFUNC5:def 12;
  end;
  now
    assume
A9: seq is convergent_to_+infty;
    then seq is convergent;
    hence lim seq = +infty by A9,MESFUNC5:def 12;
  end;
  hence thesis by A1,A7;
end;
