Suppose $f:\mathbb{R}^2\to\mathbb{R}^2$ is differentiable with uniformly bounded partial derivatives $$\left|\dfrac{\partial f_i}{\partial x_j}(x_1,x_2)\right|\leq 1$$ for all $(x_1,x_2)\in\mathbb{R}^2$ and for $i,j=1,2$. Show that there is a constant $L$ such that $\left\|f(y)-f(x) \right\|\leq L\left\|y-x\right\|$ for all $x,y\in\mathbb{R}^2.$
My idea is to use the Mean Value Theorem and I think that $L$ should be taken as the norm of the Jacobian but I don't know how to proceed.
Write $x = (x_1,x_2), y = (y_1,y_2)$. Then $|f_i(x_1,x_2) - f_i(y_1,x_2)| \le |x_1-y_1|$ and $|f_i(y_1,x_2)-f_i(y_1,y_2)| \le |x_2-y_2|$ for $i=1,2$ by MVT. So
$$||f(x)-f(y)|| \le ||f(x_1,x_2)-f(y_1,x_2)||+||f(y_1,x_2)-f(y_1,y_2)||$$ $$ \le \sqrt{2|x_1-y_1|^2}+\sqrt{2|x_2-y_2|^2} \le 2{||x-y||},$$ where this last inequality used that $a+b \le \sqrt{2(a^2+b^2)}.$