a and b are integers(adsbygoogle = window.adsbygoogle || []).push({});

Prove that:

2ab <= a^{2}+ b^{2}

I have tested various values for a and b and determined that the statement seems to be generally true. I'm having a hard time though constructing a formal proof.

It will not do to suppose the statement is wrong and then provide a counterexample. This would only prove that the statement is true for those particular values of a and b, and not for all values.

I learned about mathematical induction but I think that method is used to prove statements about a well-ordered set. a and b can be any integer so I don't think that would work.

I wondered what kind of statement may have simplified to produce a^{2}+ b^{2}so I played with (a+b) * (a+b) which produced a^{2}+ b^{2}+ 2ab and also with (a-b) * (a-b) which produced a^{2}+ b^{2}- 2ab. This looks similar to the original statement, and I feel like I may be onto some clues.

Without giving me the answer, I wonder if someone could give me a push in the right direction?

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# B How to construct a proof?

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