let X be RealNormSpace; :: thesis: for rseq1 being Real_Sequence
for seq2 being sequence of X st rseq1 is summable & ex m being Nat st
for n being Nat st m <= n holds
||.(seq2 . n).|| <= rseq1 . n holds
seq2 is norm_summable

let rseq1 be Real_Sequence; :: thesis: for seq2 being sequence of X st rseq1 is summable & ex m being Nat st
for n being Nat st m <= n holds
||.(seq2 . n).|| <= rseq1 . n holds
seq2 is norm_summable

let seq2 be sequence of X; :: thesis: ( rseq1 is summable & ex m being Nat st
for n being Nat st m <= n holds
||.(seq2 . n).|| <= rseq1 . n implies seq2 is norm_summable )

assume that
A1: rseq1 is summable and
A2: ex m being Nat st
for n being Nat st m <= n holds
||.(seq2 . n).|| <= rseq1 . n ; :: thesis: seq2 is norm_summable
consider m being Nat such that
A3: for n being Nat st m <= n holds
||.(seq2 . n).|| <= rseq1 . n by A2;
A4: now :: thesis: for n being Nat holds 0 <= ||.seq2.|| . n
let n be Nat; :: thesis: 0 <= ||.seq2.|| . n
||.(seq2 . n).|| = ||.seq2.|| . n by NORMSP_0:def 4;
hence 0 <= ||.seq2.|| . n ; :: thesis: verum
end;
now :: thesis: for n being Nat st m <= n holds
||.seq2.|| . n <= rseq1 . n
let n be Nat; :: thesis: ( m <= n implies ||.seq2.|| . n <= rseq1 . n )
assume m <= n ; :: thesis: ||.seq2.|| . n <= rseq1 . n
then ||.(seq2 . n).|| <= rseq1 . n by A3;
hence ||.seq2.|| . n <= rseq1 . n by NORMSP_0:def 4; :: thesis: verum
end;
hence ||.seq2.|| is summable by ; :: according to LOPBAN_3:def 3 :: thesis: verum