S is the set of all bounded, infinite sequences (x_k)=(x_1,x_2,...) of real numbers where (x_k) is bounded if its supremum is less than infinite for every pair of such sequences x=(x_k) and y=(y_k) where d(x,y)=sup_k abs (x_k-y_k). the question says denote by B(0,x)- the closed unit ball in (S,d) centered at 0 B(o,x)-={x: d(0,x)<=1}.

1. observe by construction B(0,x)- is closed and bounded.

2. for each n in the naturals, define x^(n)=(x_n,k) by x_n,k=1 if x=n and 0 otherwise. prove that sequence x^1,x^2,x^3,... of elements in (S,d) lies in B(0,x)- and has no convergent subsequences. conclude that B(0,x)- is not sequentially compact hence by bolzano-weierstrauss thm cannot be compact.

thanks for any help.