Prove that a subset S of a normed space is bounded if and only if nXn 0 for any xn S and any scalars n0. Show that the sum of two Cauchy sequences…

Q31. Prove that a subset S of a normed space is bounded if and only if nXn 0 for any xn S and any scalars n0.Q40. Show that the sum of two Cauchy sequences is a Cauchy sequence.Q46. Give an example of a linear mapping from a normed space into a normedspace which is not continuous.