Thread: under standing metric spaces and cauchy sequences... (analysis/topology)

hmm so i have this metric defined by x,y in R^k such that d_1(x,y)=max{abs (x_j-y_j):j=1,2,...,k} and d_2(x,y)= sigma (from j=1 to k) abs (x_j-y_j). i did the first part of the question which was to verify that these are indeed metrics for R^k. the second part of the question asks to prove that d_1 and d_2 are complete. from the book definition, the metric space (S,d) is said to be complete if every cauchy sequence in S converges to some element in S. the hint given is to consider d_1(x,y)<=d(x,y)<=sqrt(k)d_1(x,y) where d is the standard metric defined by d(x,y)=sqrt (sigma from j=1 to k) (x_j-y_j)^2. can i get some help? thanks !

Originally Posted by squarerootof2

hmm so i have this metric defined by x,y in R^k such that d_1(x,y)=max{abs (x_j-y_j):j=1,2,...,k} and d_2(x,y)= sigma (from j=1 to k) abs (x_j-y_j). i did the first part of the question which was to verify that these are indeed metrics for R^k. the second part of the question asks to prove that d_1 and d_2 are complete. from the book definition, the metric space (S,d) is said to be complete if every cauchy sequence in S converges to some element in S. the hint given is to consider d_1(x,y)<=d(x,y)<=sqrt(k)d_1(x,y) where d is the standard metric defined by d(x,y)=sqrt (sigma from j=1 to k) (x_j-y_j)^2. can i get some help? thanks !

1) is complete. Let be a Cauchy sequence, so it means, whenever . Thus, . This means that, . Thus, we see that if is a Cauchy with respect to then it is Cauchy with respect to the ordinary Euclidean metric. But Euclidean metric spaces are complete, thus is also complete.