Let d and d be two distances on the same set E so that the following is satisfied:

If U E is open with the distance d, then it is also open with the distance d.

1) Let x_nbe a sequence in E that converges with the distance d. Prove that x_nalso converges with the distance d.

2) Show an example of E, d, d and x_nthat satisfy the condition in italic letters, but x_nconverges with the distance d but not with the distance d.