SPOILER below ... On 10/27/16, Dan Asimov <asimov@msri.org> wrote:
Puzzle: Define a grunch to be any closed and bounded subset of the real line R consisting entirely of rational numbers.
True or false: There is a countable set of grunches such that every grunch is a subset of one of them.
Prove your answer.
—Dan _______________________________________________
A bounded closed subset of reals is compact, so a grunch contains all its accumulation points, which must also be rational; conversely any bounded set of rationals containing its rational accumulation points is a grunch. Suppose [H_i] is some denumerable sequence of grunches such that for each grunch S there exists i such that S is a subset of H_i . Let G = {0} union {x_i} , where x_i is rational, x_i notin Union_{j <= i} H_j , and 0 < x_i < x_{i-1} . G contains its sole accumulation point 0 , so G is a grunch; however for all i , G notin H_i --- contradiction. Hence there is no such sequence [H_i] . QED Fred Lunnon