Prove a consequence first

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Prove a consequence first

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[QUICK DESCRIPTION]

If one wants to show "If A, then B", first find an interesting consequence C of B that looks easier to prove than B itself, but not so simple that it can be immediately deduced from A. Then try to prove "If A, then C". Finally, show "If A and C, then B". The point is that this factors the original problem into two simpler ones. If one believes that the original implication is true, then the two sub-implications must be true also, so one is not "losing" anything by trying this method.

A variant approach, once one has successfully obtained "If A, then C", is to deconstruct that proof in order to find out what made it work; this can then give valuable clues as to how to then prove the stronger statement "If A, then B". This tends to work well when C acts as a "simplified toy model" of B.

[PREREQUISITES]

[EXAMPLE]

For instance, if trying to prove an identity of the form $f(n) = c$, where $f(n)$ is some expression depending on a parameter $n$, and $c$ is independent of $n$, one might first try to establish the simpler statement "$f(n)$ is independent of $n$", and then to establish $f(n_0) = c$ for some special value $n_0$ of $n$ (note that $n_0$ could be an asymptotic value, such as infinity).

Note that many proofs of identities by induction are basically of this form.

[GENERAL DISCUSSION]

This post is based on [http://www.google.com/buzz/114134834346472219368/iWPJy7vqQ7s/A-basic-problem-solving-technique-in-mathematics this Google Buzz article by Terence Tao].

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