How to proof Inequalities

Let $a,b\in\mathbb{R}$. Prove th inequality $\frac{a}{b}+\frac{b}{a}\geq{2}$

Proof

We are given $\frac{a}{b}+\frac{b}{a}\geq{2}$, .

Let $(a-b)^2\geq{0}$ so that $a^2-2ab+b^2\geq{0}\Leftrightarrow{a^2+b^2}\geq{2ab}$

$\Leftrightarrow\frac{a^2+b^2}{ab}\geq{2}$

$\Leftrightarrow\frac{a}{b}+\frac{b}{a}\geq{2}$

Equality holds if and only if $a-b=0$ or $a=b$.