$\frac{a}{b} + \frac{b}{c} + \frac{c}{a} \geq \frac{b}{a} + \frac{a}{c} + \frac{c}{b}$
$\Leftrightarrow \frac{a}{b} - \frac{b}{a} + \frac{b}{c} - \frac{c}{b} + \frac{c}{a} - \frac{a}{c} \geq 0$
$\Leftrightarrow \frac{a^{2} - b^{2}}{ab} + \frac{b^{2} - c^{2}}{bc} + \frac{c^{2} - a^{2}}{ac} \geq 0$
$\Leftrightarrow \frac{c(a^{2} - b^{2})}{abc} + \frac{a(b^{2} - c^{2})}{abc} + \frac{b(c^{2} - a^{2})}{abc} \geq 0$
$\Leftrightarrow a^{2}c - c^{2}a + c^{2}b - b^{2}c + b^{2}a - a^{2}b \geq 0$
$\Leftrightarrow ac(a - c) + bc(c - b) + ab(b - a) \geq 0$ $\Leftrightarrow ac(a - c) + bc(c - a + a - b) + ab(b - a) \geq 0$
$\Leftrightarrow ac(a - c) + bc(c - a) + bc(a - b) + ab(b - a) \geq 0$
$\Leftrightarrow (a - c)(a - b)c + (a - b)(c - a)b \geq 0$
$\Leftrightarrow (a - b)(c - a)(b - c) \geq 0$ luôn đúng với $a \geq b \geq c > 0$
