<s>[INST] Rational numbers inequality proof - is this proof valid? 
Knowing that $x^2+y^2 = 2 $ prove that $x+y \le 2$

I rewrite this as
$$x^2+y^2 \ge x+y$$
Now, multiply both sides by 2
$$x^2 -2x + 1 + y^2-2y + 1 -2 +x^2 + y^2 \ge 0$$
I substitue $2$ for $x^2+y^2$
$$(x-1)^2+(y-1)^2\ge0$$
Which is true
 [/INST] Or (shortened, once we know how to poceed):
$$x+y\le x+y+\tfrac{ (x-1)^2+(y-1)^2}2=x+y+\tfrac{x^2-2x+1+y^2-2y+1}2=\frac{x^2+y^2+2}2=2$$
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