13. Consider the notion of Cauchy sequence under di¤erent metrics.

(a) Prove proposition 5.27 in the form: In a metric space X under two equivalent metrics, d1 and d2, a sequence fxngHX is a Cauchy sequence in ðX; d1Þ i¤ fxng is a Cauchy sequence in ðX; d2Þ.

(b) Give an example of a metric on Rn,

d, so that sequences that are Cauchy under d are di¤erent than sequences that are Cauchy under the standard metric. (Hint: Consider a nonequivalent metric, like d in exercise 18 in chapter 3:Þ