Here is a more detailed ¡­

Here is a more detailed hint to the additional exercise: take a Cauchy sequence {Zn} in X/Y, thus Zn=xn+Y. First choose a subsequence {Zn'} with ||Zk'-Z{k+1}'|| smaller than 1/(2^k). Then use the fact that the definition of the norm on X/Y is an infimum to choose a Cauchy sequence in X: for k=1 find y1 such that the norm on X of x1-x2-y1 is less than 1/2 + 1/2, let X1=x1 and X2=x2+y1. Then at k=2 choose X3 and by induction get {X_k} a Cauchy sequence in X. Use its limit to get a limit in X/Y for the original Cauchy sequence.