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Let $T$ be a nonempty subset of the interval $(\,0, 1)$. If every finite subset $\{x_1,x_2,…,x_n\}$ of $T$ (with no two of equal) has the property that $x_1^2+x_2^2+⋯+x_n^2<1$, then prove that $T$ is a countable set.

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You can notice that the number of $x\in T$ which satisfy $x>a$ are finite for any $a>0$. Hence $T$ willl be at most countable.

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