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I have to teach inequality proofs and am looking for an opinion on something.

Lets say I have to prove that a^{2}+b^{2}≥2ab. (a very simple example, but I just want to demonstrate the logic behind the proof that I am questioning)

Now the correct response would be to start with the inequality (a - b)^{2}≥0, then progress to:

a^{2}+b^{2}-2ab≥0

∴ a^{2}+b^{2}≥2ab.

What many students do, as it is generally much easier, is to start with the required result and work backwards:

i.e.

a^{2}+b^{2}≥2ab

a^{2}+b^{2}-2ab≥0

∴ (a - b)^{2}≥0, which is true, ∴ the initial result must also be true.

Can anyone provide an example where starting with the result and working toward a true statement will not work? My colleagues would generally discourage this approach but it would be much more convincing to students if I could show them a situation where it won't work.

Thanks for the help guys!

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# Looking for counterexample in inequality proof

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