Lower bound for cyclic expression in four variables We are given four positive real numbers $a,b,c,d$. I would like to prove that
$$\frac ab+\frac bc+\frac cd+\frac da\ge4$$
To start solving this I assumed
$$a\ge b\ge c\ge d$$
Therefore
$$\frac ab,\frac bc,\frac cd\ge 1,\ \frac da\le 1$$
and the last one is causing the problem.

As lab bhattacharjee commented, You can use the AM-GM inequality. By applying it here, we can obtain:
$$
\begin{align*}
\frac{\frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a}}{4} & \ge \sqrt[4]{\frac{abcd}{bcda}}\\
& = \sqrt[4]{1}\\
& = 1
\end{align*}
$$
Therefore:
$$
\frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a} \ge 4
$$
