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

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Problem 1Edit

First, we wish to show that  . Let  . Then   or  .

Case 1:  

  so that  
  so that  
  and   so that  

Case 2:  

 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 3Edit

  1. Because the question asks about the square of a number, you can substitute the definition of an odd number 2n + 1 into the number to be squared. So, say x is that number, then


  2. Multiply both factors together


  3. Factor out a two for the first two terms


  4. The factor   will always be a natural number. As such, it fits the definition of an odd number, 2n + 1
Problem solved!