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Suppose that for each $\lambda$ in a set $\Lambda$ we have a positive real number $a_\lambda > 0$. Suppose also that for any natural number $n$ and any $\lambda_1, \cdots, \lambda_n \in \Lambda$ we have $ \sum_{ i = 1 }^n a_{ \lambda_i } < 1 $ Prove that the set $\Lambda$ is at most countable.

I know that every $0 < a_{ \lambda_i } < 1$. I was wondering if I could get a hint.

Thanks!

1 Answers 1

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Hint: How many $\lambda_i$ can there be in the interval $[\frac12, 1)$? How many in the interval $[\frac13, \frac12)$? $[\frac14, \frac13)$? In general, how many can there be in the interval $[\frac1{n+1},\frac1n)$?

What happens when you take the union of all those intervals and add all those numbers up?