Reference Request-Metric Spaces I am studying metric spaces for first time. We are asked to follow N. L. carothers. 
The problems are quite good but I am unable to verify my solutions.
I would like if someone can suggest me book for Metric Spaces which has Problems and solutions in detail. so that I can get confident with these topics.
Thank You
 A: Try 

Topology of Metric Spaces by S. Kumaresan 

The author motivate the concept in a geometrical way and gives lot of problems with sufficient hints! 
Added: Here's the advice of the author John M.Lee from his preface of the book "Introduction to smooth manifolds":

I have deliberately not provided written solutions to any of the problems, either
  in the back of the book or on the Internet. In my experience, if written solutions
  to problems are available, even the most conscientious students find it very hard
  to resist the temptation to look at the solutions as soon as they get stuck. But it is
  exactly at that stage of being stuck that students learn most effectively, by struggling
  to get unstuck and eventually finding a path through the thicket. Reading someone
  else’s solution too early can give one a comforting, but ultimately misleading, sense
  of understanding.
If you really feel you have run out of ideas, talk with an instructor,
  a fellow student, or one of the online mathematical discussion communities such as math.stackexchange.com. 
Even if someone else gives you a suggestion that turns out
  to be the key to getting unstuck, you will still learn much more from absorbing the
  suggestion and working out the details on your own than you would from reading
  someone else’s polished proof.

