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Math Amateur

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I need some help/clarification with Conway's defintion of completeness of a metric space ...

Conway's definition of a Cauchy sequence and a complete metric space read as follows ... ... View attachment 7637In the above text from Conway we read the following:

"... ... The discrete metric space \(\displaystyle (X,d)\) is said to be complete if every Cauchy sequence converges. ... ... My question is as follows:

Why is Conway restricting this definition to a discrete metric space ... indeed is this a misprint ... ?

Surely we can say that an arbitrary metric space \(\displaystyle (X,d)\) is said to be complete if every Cauchy sequence converges. ... ... Hope someone can help ...

Peter

NOTE: at the beginning of Section 5.2 \(\displaystyle (X,d)\) is declared to be a given (arbitrary) metric space ... as follows:View attachment 7638