Let B be a set of positive real numbers with the property that adding together any...

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Let B be a set of positive real numbers with the property that adding together any finite subsets of elements from B always gives a sum of 2 or less. Prove that B must be finite or countable. (Suggestion: Argue that B is a countable union of sets B,, defined for each n E N as the intersection of B and (1) For each n, find a (finite) bound on the cardinality of B What can we say about the countability of this union?) Let B be a set of positive real numbers with the property that adding together any finite subsets of elements from B always gives a sum of 2 or less. Prove that B must be finite or countable. (Suggestion: Argue that B is a countable union of sets B,, defined for each n E N as the intersection of B and (1) For each n, find a (finite) bound on the cardinality of B What can we say about the countability of this union?)

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For each n N Bn x B x 2n Note thatB Bn where n N Now we will show that eithe... View the full answer