 reserve Rseq, Rseq1, Rseq2 for Function of [:NAT,NAT:],REAL;

theorem
  Rseq1 is nonnegative-yielding &
  (for n,m being Nat holds Rseq1.(n,m) <= Rseq2.(n,m))
& Partial_Sums Rseq2 is P-convergent
 implies Partial_Sums Rseq1 is P-convergent
proof
   set RPS1 = Partial_Sums Rseq1;
   set RPS2 = Partial_Sums Rseq2;
   assume that
a2: Rseq1 is nonnegative-yielding &
    for n,m be Nat holds Rseq1.(n,m) <= Rseq2.(n,m) and
a1: RPS2 is P-convergent;
   for n,m be Nat holds 0 <= Rseq2.(n,m)
   proof
    let n,m be Nat;
    0 <= Rseq1.(n,m) & Rseq1.(n,m) <= Rseq2.(n,m) by a2;
    hence 0 <= Rseq2.(n,m);
   end; then
   Rseq2 is nonnegative-yielding; then
   RPS2 is non-decreasing by th80a; then
   RPS2 is bounded_above by a1; then
   consider M be Real such that
a3: M is UpperBound of RPS2.: [:NAT,NAT:] by XXREAL_2:def 10;
   now let a be ExtReal;
    assume a in RPS1.: [:NAT,NAT:]; then
    consider x be Element of [:NAT,NAT:] such that
a5:  x in [:NAT,NAT:] & a = RPS1.x by FUNCT_2:65;
    consider n,m be object such that
a6:  n in NAT & m in NAT & x = [n,m] by ZFMISC_1:def 2;
    reconsider n,m as Element of NAT by a6;
a7: RPS1.(n,m) <= RPS2.(n,m) by a2,lm84;
    RPS2.(n,m) <= M by a3,XXREAL_2:def 1,a6,FUNCT_2:35;
    hence a <= M by a5,a6,a7,XXREAL_0:2;
   end; then
   M is UpperBound of RPS1.: [:NAT,NAT:] by XXREAL_2:def 1; then
a8:RPS1 is bounded_above by XXREAL_2:def 10;
   RPS1 is non-decreasing by a2,th80a;
   hence RPS1 is P-convergent by a8;
end;
