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

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< Mathematical Proof | Appendix | Answer Key

Problem 1.1) 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 that

case 1b:

We can't actually say anything we want to with just this, so we have to also add in

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--