AM-GM inequality: $\frac{a^2}{b} + \frac{b^2}{c} + \frac{c^2}{d} + \frac{d^2}{a} \geq a + b + c + d$ 
Let $a, b, c, d > 0$. Prove that $\frac{a^2}{b} + \frac{b^2}{c} + \frac{c^2}{d} + \frac{d^2}{a} \geq a + b + c + d$.

I'm supposed to prove this by AM-GM, but I can't see how. Any help would be appreciated.

By AM-GM $\frac{a^2}{b}+b\geq2a$.
Thus, $\sum\limits_{cyc}\frac{a^2}{b}\geq\sum\limits_{cyc}(2a-b)=\sum\limits_{cyc}a$ and we are done!
