Question

Show that the sum of any real positive quantity and its reciprocal is never less than $2$.

Answer 1

Let the positive real number be $a$ then it's reciprocal will be $\frac{1}{a}$.

So using property $AM\ge GM$ we could say that

$\frac{a+\frac{1}{a}}{2}\ge \sqrt{a\times \frac{1}{a}}$ or

$(a+\frac{1}{a})\ge 2\times 1$

Therefore $(a+\frac{1}{a})\ge 2$.