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In my previous entry, I said that there was a non trivial metric on the set of finite sequences of digits which made K a continuous function. This is true, but I realised yesterday that I had a problem showing (in my mind) that the metric that I defined checked the triangular inequality.

So just for the record:

Given x = (x_{0},...,x_{n_{1}}) and y = (y_{0},...,y_{n_{2}}), with n_{1} and n_{2} integers, and assuming n_{1} <= n_{2} (note that we do not assume that x and y have go the same length) we define the metric by

d(x,y) = ( \sum_{i=0}^{n_{1}} \frac{1}{2^{i}} |x_{i}-y_{i}| ) + \frac{1}{2^{n_{1}}}

I took a picture of a little part of the proof that d is a metric (that part of the triangular inequality wasn't obvious at first, so I had to write it down)