First, choose a point in M and define
You might need to mark some points in M (to figure out the points indeed converge) and make sure each where .
Since is a contractive function, is a Cauchy sequence. Since M is a complete metric space, the sequence has a limit in M and we call it x. A contractive function in a metric space is a continuous, so converges. We know that is simply whose limit is x. Thus .
To show the uniqueness, suppose on the contrary that you have another point . Now you can draw a contradiction if you check your distance function formula