
theorem Th63:
  for L,K be ExtREAL_sequence, c be Real st 0 <=c & (for n,m be
  Nat st n <=m holds L.n <= L.m) & (for n be Nat holds K.n = c * L.n) & L
  is without-infty holds (for n,m be Nat st n <=m holds K.n <= K.m) & K is
  without-infty & K is convergent & lim K = sup rng K & lim K = c * lim L
proof
  let L,K be ExtREAL_sequence, c be Real;
  assume that
A1: 0 <= c and
A2: for n,m be Nat st n <=m holds L.n <= L.m and
A3: for n be Nat holds K.n = c * L.n and
A4: L is without-infty;
A5: sup rng L = lim L by A2,Th54;
  thus
A6: now
    let n,m be Nat;
    assume n <= m;
    then c * L.n <= c * L.m by A1,A2,XXREAL_3:71;
    then K.n <= c * L.m by A3;
    hence K.n <= K.m by A3;
  end;
  thus K is without-infty by A1,A3,A4,Th62;
  thus K is convergent & lim K= sup rng K by A6,Th54;
  sup rng K = lim K by A6,Th54;
  hence thesis by A1,A3,A4,A5,Th62;
end;
