<s>[INST] How can I prove that $2/9$x$ and $y$ are real numbers.
Given that $1<x^2-xy+y^2<2$, how can I show that $\frac 29<x^4+y^4<8$ ?
Then can I use that to prove that for any natural number $n>3$
$$x^{2^n}+y^{2^n}>\frac 2{3^{2^n}} \text{?}$$
 [/INST] $0\leq x^2+y^2<2+xy$. Squaring it, you get $x^4+y^4+2x^2y^2<4+4xy+x^2y^2\Rightarrow x^4+y^4<4+4xy-x^2y^2=8-(4+x^2y^2-4xy)=8-(2-xy)^2\leq 8$
</s>