Let {qn} be an enumeration of the rationals, and for each n, let Un be an open interval of length 1/2n centered at qn. Denote the union of all Un by U. Then U is a dense open . . . [Read More]
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To formalize . . . [Read More] Let k ≥ 0, and let n = k(2k + 1). Then we have: n2 + (n+1)2 + … + (n+k)2 = (n+k+1)2 + … + (n+2k)2. For example, |
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This is a classic introductory result in combinatorics. Suppose that six people are gathered at a dinner party. Then there is a group of three people at the party who are either all mutual acquaintances or all mutual strangers.