1. (5 points) Let X, || . || be a normed vector space. Give careful, precise definitions for each of the fol- lowing. a. The sequence In converges to x in the space X b. In is a Cauchy sequence in the space X c. The series In is convergent in the space X n=0 d. The series In is absolutely convergent in the space X n=0 e. The space X is complete2. (5 points) Given sequences of real numbers an, bn, and In such that an _ In _ bn for all n 2 1, prove that if lim an = lim bn = L n-too then lim In = L Your proof must provide a careful, precise verification according to your definition in la. above, including specific choices for X and || . || appropriate to this