<s>[INST] Prove $\lim\limits_{x^2 + y^2 \to +\infty} x^2 -2xy + 2y^2 = +\infty$ 
Prove that $$\lim_{x^2 + y^2 \to +\infty} x^2 -2xy + 2y^2 = +\infty$$


My attempt:
$$x^2 + 2y^2 = x^2+y^2 + y^2  \implies \lim_{x^2 + y^2 \to +\infty}x^2 +2y^2 = +\infty$$
Then, from Cauchy-Schwarz:
$$x^2 + 2y^2 \geq 2\sqrt2xy \geq 2xy $$
Thus,
$$x^2+2y^2 -2xy \geq 0$$
I think I am on the correct path, but I don't know how to proceed.
 [/INST] Because $$x^2-2xy+2y^2\geq\frac{1}{3}(x^2+y^2)\rightarrow+\infty.$$
$$x^2-2xy+2y^2\geq\frac{1}{3}(x^2+y^2)$$ it's
$$3x^2-6xy+6y^2\geq x^2+y^2$$ or
$$2x^2-6xy+5y^2\geq0,$$ which is true because $$3^2-2\cdot5<0.$$
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