Originally Posted by

**kingwinner** Let

d(m,n)= 1/2 + ∑1/(2^k) where the sum is from k=m to k=n-1, if m<n

d(m,n)=0, if m=n

d(m,n)=d(n,m), if m>n

d(1,2)=1, d(2,3)=0.75, d(3,4)=0.625

Define

B1=Closed ball of radius 0.1 about 2={1,2,3,4,...}

B2=Closed ball of radius 0.75 about 3={2,3,4,...}

B3=Closed ball of radius 0.625 about 3={3,4,5,...} <---Is this correct? Could someone confirm this?

...

If so, then it's a decreasing sequence of closed balls with empty intersection.

Now the problem is whether this metric space (N,d) is complete or not. (complete means every Cauchy sequence in N converges (in N))

For any n E N, the open ball of radius 1/2 about n = B(1/2,n) = {n}. But does this show that every Cauchy sequence in N converges (in N)) ??? I'm puzzled about this part, and I would appreciate if someone can help me with this part.

Thanks for any help!