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To prove $x+\frac{1}{x}\geq2$ where $x$ is a positive real number.
This is what i try:
$$\text{We need to prove } \hspace{1cm} x+\frac{1}{x}-2\geq 0$$
now,$$\frac{x^2-2x+1}{x}=(x-2)+\frac{1}{x}$$
its enough to show that $$\frac{1}{x}\geq(x-2)\hspace{0.5cm} \text{ when } \hspace{0.2cm}0<x\leq 2$$
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