Let us consider three sequences $(a_n)_{n\ge1}$, $(b_n)_{n\ge1}$ and $(c_n)_{n\ge1}$ having the properties:
- $a_{n},\ b_{n},\ c_{n}\in\left(0,\ \infty\right)$
- $a_{n}+b_{n}+c_{n}\ge\frac{a_{n}}{b_{n}}+\frac{b_{n}}{c_{n}}+\frac{c_{n}}{a_{n}}\ \forall n\ge1$
- $\lim\limits_{n\to\infty}a_{n}b_{n}c_{n}=1 $
Prove that $$\lim_{n\to\infty}\frac{a_{n}+b_{n}+c_{n}}{a_{n}b_{n}+b_{n}c_{n}+c_{n}a_{n}}=1 $$