
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, . . ..)

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

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

Re: Countability problem
If $\displaystyle x\in T$, then $\displaystyle x\in (0,1)$ so $\displaystyle x\in ((k+1)^{1},k^{1})$ for some $\displaystyle k$. This gives that $\displaystyle x\in T_k$. Conversely, if $\displaystyle x$ is in a $\displaystyle T_k$, it's in $\displaystyle T$.

Re: Countability problem
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.

Re: Countability problem
What if there were infinitely many elements in $\displaystyle T_k$ (for example, more than $\displaystyle k+2$)?

Re: Countability problem
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?

Re: Countability problem

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