Mathematical Proof/Appendix/Answer Key/Mathematical Proof/Methods of Proof/Constructive Proof

__TOC__ Problem 1 First, we wish to show that . Let . Then or . Case 1: so that so that and so that Case 2: and so that so that and so that Since in both cases, , we know that Now we wish to show that . Let . Then and . Case 1a: , so Case 1b: We can't actually conclude anything we want with just this, so we have to also to consider the case . Case 2a: : [see Case 1a] Case 2b: We now have and so that Of course, since , it follows that .Since both cases 2a and 2b yield , we know that it follows from 1b. Since in both cases 1a and 1b, , we know that . Since both and , it follows (finally) that . --will continue later, feel free to refine it if you feel it can be-- Problem 3 Because the question asks about the square of a number, you can substitute the definition of an odd number 1.2em 2n + 1 into the number to be squared. So, say x is that number, then Multiply both factors together Factor out a two for the first two terms The factor will always be a natural number. As such, it fits the definition of an odd number, 1.2em 2n + 1 Problem solved!