Intuition behind the Banach fixed-point theorem The theorem appeared as an exercise in my real analysis book and only considered functions in $\mathbb{R}$ but the proof of the general theorem seems to be almost identical after looking up the wikipedia article. Is there a somewhat intuitive way of thinking about this theorem? I understand the proof and what it entails but I don't see why the result ought to hold given the necessary conditions. Is there a way of convincing someone that the theorem ought to be true without actually proving it? Some vague geometric intuition would be nice to have a picture in my head of what's going on. 

Generally, a function between metric spaces can be very wild with respect to the distances. But if we demand that the function actually decreases the distance between any two points, then it becomes difficult to actually construct such functions. Looking at a few examples of such functions one sees that one easy way to obtain such a function from a space to itself is to choose a point and treat it as a sort of magnet, where the function describes how points move toward it, as if the point exerts a gravitaional field, thus shrinking distances. If the gravitational pull is strong enough, the shrinking will be at least by a factor smaller than 1. The Banach fixed point theorem says that any endo-function shrinking distances sufficiently fast must be the result of such a point pulling things to it. If the space is not complete then that point may not be in the space but rather in its completion, but it’s still ‘there’. So, if globally the function behaves like the result of a pulling point with sufficient force, then that point is really there, since (intuitively?) if there is no point pulling everything to it, how can all the distances be tapidly shrinking no matter where you are?
