Exercise 1.5.8

Question

Let $B$ be a set of positive real numbers with the property that adding together any finite subset of elements from $B$ always gives a sum of 2 or less. Show $B$ must be finite or countable.

Ulisse Mini · Accepted Answer

Notice $B \cap (a, 2)$ is finite for all $a > 0$, since if it was infinite we could make a set with sum greater then two.
  And since $B$ is the countable union of finite sets $\bigcup_{n=1}^\infty B \cap (1/n, 2)$, $B$ must be countable or finite.

Exercise 1.5.8

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Exercise 1.5.8

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