Lipschitz condition in 2D

Question

How can I prove that

$|x_1 x_2 - y_1 y_2| \leq L\big(|x_1 - y_1| + |x_2 - y_2|\big)$

where

$x_1,y_1 \in [0,K]$

$x_2,y_2 \in [0,M]$

$K$ and $M$ are some constant

$L$ is constant from Lipschitz condition

Answer 1

Instead of using the mean value theorem, we can instead just use the triangle inequality $$\begin{align} |x_1 x_2 - y_1y_2| &\leq |x_1x_2-x_1y_2+x_1y_2-y_1y_2|\\ &\leq |x_1(x_2-y_2)+y_2(x_1-y_1)| \\ &\leq|x_1(x_2-y_2)|+|y_2(x_1-y_1)| \\ &=|x_1||x_2-y_2|+|y_2||x_1-y_1| \\ &\leq M |x_1 - y_1| + K |x_2 - y_2| \\ &\leq L \big(|x_1 - y_1| + |x_2 - y_2|\big) \end{align}$$

where $L = \max(K, M)$.

Answer 2

Let $f(x) = x_1x_2$, note that $Df(x)h = x_2 h_1 + x_1 h_2$ and so $\|Df(x)\| \le \|x\|$.

Let $L = \max_{x \in [0,K]\times [0,M]} \|x\|$, then the mean value theorem gives $|f(x)-f(y)| \le L \|x-y\|$.