let X be ComplexUnitarySpace; for seq being sequence of X
for n, m being Nat holds ||.((Sum (seq,m)) - (Sum (seq,n))).|| <= |.((Sum (||.seq.||,m)) - (Sum (||.seq.||,n))).|
let seq be sequence of X; for n, m being Nat holds ||.((Sum (seq,m)) - (Sum (seq,n))).|| <= |.((Sum (||.seq.||,m)) - (Sum (||.seq.||,n))).|
let n, m be Nat; ||.((Sum (seq,m)) - (Sum (seq,n))).|| <= |.((Sum (||.seq.||,m)) - (Sum (||.seq.||,n))).|
||.((Sum (seq,m)) - ((Partial_Sums seq) . n)).|| <= |.(((Partial_Sums ||.seq.||) . m) - ((Partial_Sums ||.seq.||) . n)).|
by Th39;
then
||.((Sum (seq,m)) - (Sum (seq,n))).|| <= |.((Sum (||.seq.||,m)) - ((Partial_Sums ||.seq.||) . n)).|
by SERIES_1:def 5;
hence
||.((Sum (seq,m)) - (Sum (seq,n))).|| <= |.((Sum (||.seq.||,m)) - (Sum (||.seq.||,n))).|
by SERIES_1:def 5; verum