Several duplicate targets have been suggested in the comments. They don't answer the whole question (surjectivity and isometry are dealt with separately), but I think closing is still appropriate.
Yes, it makes sense for users to vote as a duplicate of different questions. The OP should have isolated a specific issue with the problem...
@NajibIdrissi There is a very large number of users who study real analysis + metric spaces following Rudin's PMA or something equivalent. They tend to upvote questions they can understand and find interesting but can't answer.
