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

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Problem 1[edit]

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 3[edit]

  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!