let X be ComplexUnitarySpace; for seq being sequence of X
for n, m being Nat holds ||.(Sum (seq,m,n)).|| <= |.(Sum (||.seq.||,m,n)).|
let seq be sequence of X; for n, m being Nat holds ||.(Sum (seq,m,n)).|| <= |.(Sum (||.seq.||,m,n)).|
let n, m be Nat; ||.(Sum (seq,m,n)).|| <= |.(Sum (||.seq.||,m,n)).|
||.((Sum (seq,m)) - (Sum (seq,n))).|| <= |.((Sum (||.seq.||,m)) - (Sum (||.seq.||,n))).|
by Th40;
hence
||.(Sum (seq,m,n)).|| <= |.(Sum (||.seq.||,m,n)).|
by SERIES_1:def 6; verum