let X be ComplexUnitarySpace; :: thesis: for seq being sequence of X st X is complete & seq is Cauchy holds
seq is bounded
let seq be sequence of X; :: thesis: ( X is complete & seq is Cauchy implies seq is bounded )
assume ( X is complete & seq is Cauchy ) ; :: thesis: seq is bounded
then seq is convergent by Def12;
hence seq is bounded by Th80; :: thesis: verum