Hey Chiro,

I am looking for a way to describe sequences that are Cauchy with respect to multiple norms simultaneously. For example, I want to extend the rationals by all sequences that converge with respect to both the p-adic and q-adic norms. So, I hoped each sequence that converges with respect to both of those norms might also converge with respect to a third norm (one that somehow combines the two). But, I think an easier way of dealing with it might be to avoid norms and use a metric, instead. I can define \(\displaystyle d:\mathbb{Q} \times \mathbb{Q} \to [0,\infty)\) by \(\displaystyle d(a,b) = \max\{|b-a|_p,|b-a|_q\}\). Then, I can use that metric to evaluate sequences, and extend the rationals that way.