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

< Mathematical Proof | Appendix | Answer Key### Problem 1Edit

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 3Edit

- 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
- 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, 2n + 1

- Problem solved!