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...Spivak notes problems and solutions jackceroni contents introduction chapter chpater other proofs date october the goal of this set is to solve most challenging in write up a clean concise way i apologize advance for any possible mistakes or instances which may skip over certain important points problem prove that if f x y where then all proof we go through steps as organized clearly will have so either assume would imply n can by induction assuming noting contradiction our initial assumption first note next natural rst thus identity integers now see b it follows well dened greater than know real number r since clear from previous result there exists some such rational between two reals...