Let $a\_{i}>0,i=1,2,\cdots,n$, and $a\_{1}+a\_{2}+\cdots+a\_{n}=1$.

How can we prove that

$$\displaystyle\sum\_{i=1}^{n}\dfrac{a\_{i}}{a\_{i+1}}\ge\displaystyle\sum\_{i=1}^{n}\dfrac{1-a\_{i+1}}{1-a\_{i}}$$

where $a\_{n+1}=a\_{1}$?

I think this can be done using the AM-GM inequality.