I want to show that the projection map $\mathbb{R^2}\to\mathbb{R}, (x,y)\mapsto x$ is continuous using the standard metric for $\mathbb{R^2}$ and the standard metric for $\mathbb{R}$
Proof
Let $\epsilon>0$, $\delta_{0}=\frac{\epsilon}{2}$
Let $x, y, x_{0}, y_{0}\in\mathbb{R}$ and $\sqrt{(x-x_{0})^2+(y-y_{0})^2}<\delta_{0}$
$$|x-x_{0}|=\sqrt{(x-x_{0})^2}\le\sqrt{(x-x_{0})^2+(y-y_{0})^2}<\delta_{0}<\epsilon$$
Is my approach correct?