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Students who are beginning to learn proofs (and some seasoned pros) occasionally commit the error of assuming what they're trying to prove. My question involves assuming at the onset that a solution to a problem exists, then using that existence to get the solution you assumed existed. Two examples:

  1. A common technique for proving that any function $f$ can be written as the sum of an even and an odd function is to assume such functions $O$ and $E$ exist, then note \begin{align*} f(x) &= E(x) + O(x) \\ f(-x) &= E(x) - O(x)\end{align*} and solve for $O(x)$ and $E(x)$.

  2. A nested root such as $\sqrt{2+\sqrt{\vphantom{2^5}3}}$ can be written as $\sqrt{1/2} + \sqrt{3/2}$ by first assuming there exist $A$ and $B$ such that $\sqrt A + \sqrt B$ is equal to the original radical, then solving for $A$ and $B$.

In general, when and how is it legal to assume the solution to a problem exists a priori? Or do the ends justify the means? Or is there some higher structure where the existence of solutions to the two problems above, and others like them, are proven, but I haven't gotten there yet (similar to telling Calc II students that a partial fraction decomposition exists, now find it)?

Jon
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    In both of those examples you show what the solution must be provided that it exists, and then you verify that this actually is a solution. In some other examples that I’ve seen the steps are interchangeable: you can show in either order that a solution must exist (e.g., that a certain sequence is convergent) and what it must be if it exists (e.g., what the limit is). – Brian M. Scott Dec 08 '14 at 04:20
  • @BrianM.Scott Proving a solution exists and then finding the solution I get. The gray area is pointing at a problem and saying "If the solution exists then it would have this property, which leads to that, and ta-da! Here's the solution I assumed existed." I'm comfortable solving things in this manner, I'm just curious about justification. – Jon Dec 08 '14 at 05:20
  • But that isn't a grey area: it's simply incomplete. It only becomes a complete solution when you either prove that the tentative solution that you've obtained actually is a solution, or prove independently that there is a solution. – Brian M. Scott Dec 08 '14 at 06:36
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    Your point being, assuming the solution exists will not alter whether or not a solution exists? I guess I can buy that. – Jon Dec 08 '14 at 16:49
  • That's true, but it's not my point. My point is that you seem to be misunderstanding the arguments in question. In all of them one does prove that a solution exists, either independently of finding it, or by verifying that one's tentative solution actually is a solution. – Brian M. Scott Dec 08 '14 at 18:10
  • I guess that's my main question. For example, the fact that I even bother looking for an even and an odd function whose sum is $f$ means I have to have reason to believe those functions exist in the first place. What tells me that? – Jon Dec 08 '14 at 22:11
  • In effect you’re asking how new results are discovered. That’s far too broad a question to be answered here, but experiment, observation, insight, and mathematical intuition all play a rôle. – Brian M. Scott Dec 08 '14 at 22:15
  • No, I'm asking how this method of discovery, through assuming a solution exists beforehand, is valid. – Jon Dec 09 '14 at 01:10
  • Validity isn’t a concept that applies to methods of discovery. Suppose that you guess that some problem has a solution, assume that to be the case, and prove that the only possible solution is so-and-so. At this point you have not solved the problem; you’ve shown that if the problem has a solution, that solution must be so-and-so. You still don’t know whether the problem has a solution. Your result doesn’t become a solution of the problem until you either verify in some way that the only possible solution actually is a solution or show independently that there is a solution. And at ... – Brian M. Scott Dec 09 '14 at 01:24
  • ... this point there is nothing remotely circular in the argument. It doesn’t matter whether you guessed the solution by discovering that the solution had to be so-and-so if it existed at all, or whether the Oracle of Delphi suggested it to you; in either case you still have to prove that it is a solution. – Brian M. Scott Dec 09 '14 at 01:26
  • I call it "proof by admission" and use it to show that the function which gives the area under the curve of a function f is its primitive function, what is called the "indefinite integral" (see http://math.stackexchange.com/questions/586107/what-is-the-difference-between-an-indefinite-integral-and-an-antiderivative). – Arik Sep 04 '15 at 08:23

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