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!