MHB Contraction Map: Proving $f(x)$ is a Contraction

[onie mti]

I am given this function

$f(x)=\langle (1/9) \cos(x_1+ \sin(x_2)) , (1/6) \arctan(x_1+ x_2) \rangle$

where $x_1= \langle 0,-1 \rangle$.

may I please get hints on how to prove that this function is a contraction map

 

[Jhoneine]

?Hint: To prove that $f$ is a contraction map, you need to show that it satisfies the following property: for all $x,y \in X$, there exists some constant $k \in [0,1)$ such that $\|f(x)-f(y)\| \leq k\|x-y\|$. In other words, the distance between the images of two points must be less than or equal to $k$ times the distance between the two original points.