<s>[INST] $a_n=\min\limits_{x+y=1}(x^n+y^n)=\frac{1}{2^{n-1}}$ We must prove :$\min\limits_{x+y=1}(x^n+y^n)=\frac{1}{2^{n-1}} $ for all $n \in \mathbb{N}_{>0}$
And to prove this we can use the inequality: $\frac{x+y}{2}\leq (\frac{x^n+y^n}{2})^{\frac{1}{n}}$
where equality is satisfied if $x=y=1/2$
My question is how we can prove inquality: $\frac{x+y}{2}\leq (\frac{x^n+y^n}{2})^{\frac{1}{n}}$ ? 
 [/INST] This is just convexity of the function $f(x)=x^{n}$. The inequality simply says $f(\frac {x+y} 2) \leq \frac {f(x)+f(y)} 2$. 
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