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Discussion: This is a proposed solution to problem 2-13 in Apostol's "Mathematical Analysis". The method came to me after a lot of thought but it seems kind of bizarre and I'm wondering if there's a better way to prove this. I especially think the last part could be made more rigorous/explicit.
Also, I'm not even sure my proof is valid! Tell me what you guys think.
Also feel free to critique writing style, minor errors, choice of variable names, etc...
THEOREM: Let f be a real-valued function defined on the interval [0,1] with the following property: There exists a positive real number M such that for any finite collection \{x_1,\ldots,x_n\} of elements of [0,1], |f(x_1)+\cdots +f(x_n)|\leq M. Let S denote the set of all real numbers 0\leq x \leq 1 such that f(x)\not= 0. Then S is countable.
NOTATION: [x] denotes the greatest integer less than x. S_T denotes the set of all real numbers x in [0,1] such that f(x) \epsilon T.
PROOF: We prove the statement by contradiction. Assume S is uncountable. Then either S_{(-\infty,0)} or S_{(0,+\infty)} is uncountable (or both).
Also, I'm not even sure my proof is valid! Tell me what you guys think.
Also feel free to critique writing style, minor errors, choice of variable names, etc...
THEOREM: Let f be a real-valued function defined on the interval [0,1] with the following property: There exists a positive real number M such that for any finite collection \{x_1,\ldots,x_n\} of elements of [0,1], |f(x_1)+\cdots +f(x_n)|\leq M. Let S denote the set of all real numbers 0\leq x \leq 1 such that f(x)\not= 0. Then S is countable.
NOTATION: [x] denotes the greatest integer less than x. S_T denotes the set of all real numbers x in [0,1] such that f(x) \epsilon T.
PROOF: We prove the statement by contradiction. Assume S is uncountable. Then either S_{(-\infty,0)} or S_{(0,+\infty)} is uncountable (or both).
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