Thread: Countability problem

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, . . ..)

I guess you tried to use the hint: where are you stuck?

In fact, I'm ok with the hint part. But I can't figure out how to prove T_k as a finite set.

If , then so for some . This gives that . Conversely, if is in a , it's in .

First, really thanks for your kindness help.
What I need is the part that utilize #x1^2+x2^2+· · ·+xn^2 < 1# to prove T_k is finite set, I just can't link them up.

What if there were infinitely many elements in (for example, more than )?

Since |x| greater than or equal to 1/(k+1), there is at most (k+1)^2 elements, otherwise it'll contradict the inequality.
Is it correct?

Yes.

Haha, solved. That's great. Nice to meet you, thank you very much!