sentences:  Proof.A subset of a normed space is bounded if and only if it lies in some multiple of the unit sphere; thus formula_51 for every formula_46 if formula_53 is a scalar, then formula_54 so that 
  
  The triangle inequality in formula_34 shows that
  
  for every formula_58 with formula_59.Thus
  
  If formula_61, then formula_62 for some formula_58; hence formula_64.