How to prove this inequality $ x + \frac{1}{x} \geq 2 $ I was asked to prove that: 
$$x + \frac{1}{x}\geqslant 2$$
for all values of $ x > 0 $
I tried substituting random numbers into $x$ and I did get the answer greater than $2$. But I have a feeling that this is an unprofessional way of proving this. So how do I prove this inequality?

Hint: $x^2 -2x + 1 = (x-1)^2 \ge 0$. If you do not see it, divide the inequality through by $x$.
