Concrete Example of the Contraction Principle I'm preparing for an exam, and I had trouble with another problem.
The problem asks:


Show that the system of equations
$x_1 + \frac{1}{10}cos(sin(2x_2+x_1)) = 6$
$x_2 + \frac{1}{12}e^{-{x_1}^2} + \frac{1}{10}cos(x_1+x_2) = 7$
have an unique solution $(x_1,x_2)$ in $\mathbb{R}^2$


I believe this translates to showing that the linear transformation


$T\begin{bmatrix} x_1 \\ x_2 \\ \end{bmatrix} = \begin{bmatrix} 6 - \frac{1}{10}cos(sin(2x_2+x_1)) \\ 7 - \frac{1}{12}e^{-{x_1}^2} - \frac{1}{10}cos(x_1+x_2) \end{bmatrix}$


is a contraction.  However, this is what I have trouble with; namely, those constants $6$ and $7$.  Am I on the correct track here? I still have yet to really get a grasp on estimation techniques required in analysis.

The constants 6 and 7 will disappear once you consider the difference $\|f(x)-f(y)\|$ for arbitrary $x,y\in\mathbb{R}^2$, where $f$ is the right-hand-side of your equality. 
