Hi,
I need help to prove that the space of sequences of natural numbers X={xk} endowed with the following metric is complete:
d(X,Y)=1/min{k:xk is different from yk}
Thanks in advance
Good. Now start by looking at simple cases. In the case that {x}= {1, 3, 2, 4 ....} and {y}= {1, 3, 4, 3, ...} d(x,y)= 1/3 because the third place is the first in which they are different.
What would it mean it $\displaystyle d(x, y)= 0$?
Notice that here the members of the metric space are sequences of real numbers so the "sequences" referred to in that definition are sequence of sequences!
If x and y are such sequences of different sequences, and d(x, y) goes to 0 what does that tell you about the sequences in x and y?