Let (S,d) be the metric space of all sequences, S=CNorRN where

d(x,y):=n=12n1min(1,pn(xy)),

pn(x):=max1knxk,x,yS

Show thatis a complete metric space.

a) First show thatxnxkN,xknxk,n.

b) ii) Then show completion.

Note: A complete metric space has the property that all fundamental (Cauchy) sequences converge in the same space.