L^p Distances
The L^p norms are a family of norms on vector spaces that are a generalization of Euclidean distance. Roughly || x ||_p =(x_1^p+...+x_n^p)^(1/p). The norm gives us a metric by taking the norm of the difference between 2 vectors. (That metric gives a complete topology.)
With this sketch, you can play around with these L^p distances. A standard question is 'what does a circle look like?' or you can add a third point and investigate constant sums and differences.
Would the limit of these distance functions be a distance function? What would it be like?