2004-5.md

course	course_year	question_number	tags	title	year
Analysis II	IB	5	IB 2004 Analysis II	$3 . \mathrm{II} . 16 \mathrm{~F} \quad$	2004

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 $f$.