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. 

One could image a map of let's say the state Wyoming lying somewhere in Wyoming (it looks nicely square-shaped, but you could choose any country/state). The Banach fixed point theorem then says that exactly one point on this map lies exactly on the corresponding point in the state.
The state is non-empty obviously, the metric is the Euclidean metric in $\mathbb{R}^2$, the state is obviously bounded (not really needed here), and we assume that the boundaries of the state are part of the state, so it is a closed space (I mean closed as a subset of the complete space $\mathbb{R}^2$), and therefore complete. Since Wyoming is our chosen space, the map must lie fully in this space (so not half outside Wyoming). Our contracting mapping is just the scale of the map. Now we have that the Banach fixed point theorem holds.
Let's say the state were an open set (so the space is not complete), then we could place the map such that the left boundary of the map lies on the left boundary of the state. Then there is no fixed point (the fixed point would be on the boundary, but that is not in the space now).
