I guess you tried to use the hint: where are you stuck?
I've tried for a long time, still can't figure it out. Please help me, thanks.
Let T be a nonempty subset of (−1, 0)union(0, 1).
If every finite subset {x1, x2, . . . ,xn} of T (with no two of x1, x2, . . . ,xn equal) has the property that x1^2+x2^2+· · ·+xn^2 < 1, then prove that T is a countable set.
(Hint: For every positive integer k, is the set T_k = {x : x belongs to T and |x| belongs to [1/(k+1),1/k)} a finite set? What is the union of [1/(k+1), 1/k) for k = 1, 2, 3, . . ..)