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show that for any integers x and y : 2xy <= x^2 +y^2
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Start from the inequality
>= 0,
which is valid for all real numbers "x" and "y".
It is equivalent to
>= 0.
Add 2xy to both sides. You will get an equivalent inequality
>= 2xy.
It is EXACTLY what has to be proved.
Solved and completed.
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It is one of the classic and basic inequalities in Elementary Math, but the problem's formulation is NOT ADEQUATE.
The adequate formulation is THIS :
show that for any real numbers x and y 2xy <= x^2 +y^2.
with replacing "any integers x and y" by "any real numbers x and y".
Actually, this replacement is VERY important and shows that the person who invented/created the original post,
is UNPROFESSIONAL in Math.