21127 Study Guide - Quiz Guide: Mathematical Induction, Holda, Burh
16 Dec 2015
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Here is a fact that we all know from high school (or middle school, even!) but we should prove to make sure we know it. An odd integer times an odd integer is also an odd integer. (note: a lemma is like a mini-theorem , a basic fact that follows from de nitions and fundamental results that we can use to prove more important things later. It acts like a building block; we don"t want to have to prove it every time we use it, so let"s just prove it once and get the conclusion. ) To prove this, we rst need an actual de nition of what it means to be an odd integer! Let x be an arbitrary and xed integer. We say that x is odd if and only if we can nd an integer k such that x = 2k + 1. A proof of this fact must appeal to the de nition.
