<s>[INST] Algebraic inequality:$\frac a{b^2} + \frac b{c^2} +\frac c{a^2} \geq \frac1a+\frac1b+ \frac1c$ Prove that : 
If a,b,c $\in \mathbb{R^+}$
$$\frac a{b^2} + \frac b{c^2} +\frac c{a^2} \geq \frac1a+\frac1b+ \frac1c$$
My attempt : 
We know that the sequence {a,b,c} and {$\frac1{a^2},\frac1{b^2},\frac1{c^2}$} are oppositely ordered thus from rearrenegement inequality we conclude - 
$$\frac a{b^2} + \frac b{c^2} +\frac c{a^2} \geq \frac{a}{a^2}+\frac{b}{b^2}+ \frac{c}{c^2}$$
Is this correct?
 [/INST] $\left(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\right)\left(\dfrac{a}{b^{2}}+\dfrac{b}{c^{2}}+\dfrac{c}{a^{2}}\right)\geq\left(\dfrac{1}{b}+\dfrac{1}{c}+\dfrac{1}{a}\right)^{2}$ : Cauchy-Schwarz    
$\therefore$ $\dfrac{a}{b^{2}}+\dfrac{b}{c^{2}}+\dfrac{c}{a^{2}}\geq \dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}$
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