# Talk:Combination Theorem for Cauchy Sequences/Product Rule

I was hoping to use this theorem and the theorem Combination Theorem for Cauchy Sequences/Sum Rule to establish that the Cauchy Sequences of a valued field are a commutative ring with pointwise addition and multiplication, and thereby establish that the Completion of Valued Field was a field. But I no longer think that this will do it as the theorems are about normed vector spaces and not normed division rings. What do others think? --Leigh.Samphier (talk) 07:07, 29 August 2018 (EDT)

If the normed etc. etc. can all be proved to be metric spaces, then the job is done because the results have already been proved for metric spaces. --prime mover (talk) 09:42, 29 August 2018 (EDT)