<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] We have that
$$x^2 -2xy + 2y^2 =(x-y)^2+y^2\ge y^2$$
$$x^2 -2xy + 2y^2 =\left(\frac{x}{\sqrt 2}-\sqrt 2y\right)^2+\frac{x^2}2\ge \frac{x^2}2 $$
therefore
$$x^2 -2xy + 2y^2 \ge \frac{x^2+2y^2}4 \ge \frac{x^2+y^2}4\to \infty$$
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