Product of two Hölder continuous function on a bounded interval is also Hölder continuous. Prove that the product of two Hölder-1/2 continuous functions on a bounded interval is also Hölder-1/2 continuous.

Given a function on a bounded interval, $f:I\rightarrow \mathbb{R}$,  $f$ is Hölder-1/2 continuous if there exists some $M$ such that $|f(x_1)-f(x_2)|\leq M|x_1-x_2|^{1/2}$. But I'm having trouble how to link this to the product of two Hölder continuous functions, $fg:I\rightarrow\mathbb{R}$.
Hint:
$$f(x_1)g(x_1)-f(x_2)g(x_2)=\\ 
=\left[f(x_1)-f(x_2)\right]\cdot \left[g(x_1)-g(x_2)\right]+f(x_2)\cdot\left[g(x_1)-g(x_2)\right]+g(x_2)\cdot \left[f(x_1)-f(x_2)\right]$$
Hint:
$$f(x_1)g(x_1)-f(x_2)g(x_2)=\\ 
=\left[f(x_1)-f(x_2)\right]\cdot \left[g(x_1)-g(x_2)\right]+f(x_2)\cdot\left[g(x_1)-g(x_2)\right]+g(x_2)\cdot \left[f(x_1)-f(x_2)\right]$$