How can we prove that the projection map $p\colon\mathbb{R}^2\to\mathbb{R}$, $p(x,y)=x$, is continuous?
This is a very simple question, but I have only had to prove continuity from $\mathbb{R} \to \mathbb{R}$ before, so I would greatly appreciate it if someone could help me how to do this correctly, can we do it using the open balls
Where $c=(x_1,x_2)\in \mathbb{R^{2}}$
$B_{\delta}(c)=\{(x,y): \| (x,y)-c \|<\delta\}$.
$B_{\epsilon}(p(c))=\{x:|x-p(c)|<\epsilon\}$.
And show $\forall c \in \mathbb{R^{2}}$ if $a\in B_{\delta}(c) \implies p(a)\in B_{\epsilon}(p(c)) $.