
theorem
  for X be RealNormSpace for rseq1 be Real_Sequence for seq2 be sequence
  of X holds rseq1 is summable & (ex m be Nat st
   for n be Nat st m<=n holds ||.seq2.n.|| <= rseq1.n )
    implies seq2 is norm_summable
proof
  let X be RealNormSpace;
  let rseq1 be Real_Sequence;
  let seq2 be sequence of X;
  assume that
A1: rseq1 is summable and
A2: ex m be Nat st for n be Nat st m<=n holds ||.
  seq2.n.|| <= rseq1.n;
  consider m be Nat such that
A3: for n be Nat st m<=n holds ||.seq2.n.|| <= rseq1.n by A2;
A4: now
    let n be Nat;
    ||.seq2.n.||=||.seq2.||.n by NORMSP_0:def 4;
    hence 0 <= ||.seq2.||.n;
  end;
  now
    let n be Nat;
    assume m <= n;
    then ||.seq2.n.|| <= rseq1.n by A3;
    hence ||.seq2.||.n <= rseq1.n by NORMSP_0:def 4;
  end;
  hence ||.seq2.|| is summable by A1,A4,SERIES_1:19;
end;
