State and prove the contraction mapping theorem.

Let $a$ be a positive real number, and take $X=\left[\sqrt{\frac{a}{2}}, \infty\right)$. Prove that the function

$$f(x)=\frac{1}{2}\left(x+\frac{a}{x}\right)$$

is a contraction from $X$ to $X$. Find the unique fixed point of $f$.