Centering charts about a point in a euclidean space.

Question

Let $M$ be a locally euclidean space. Let $p\in M$. Then there exists some chart $(V,\psi)$ such that $\psi(p)=0$.

Now, I first thought of breaking it into cases, but I didn't like that approach, so I was able to think of another way:

By definition of $M$, $p$ is in the coordinate domain of some chart, $(U,\phi)$ . Define another map $g:U\rightarrow \mathbb{R}^n$ by $g(x)=\phi(x)-\phi(p)$ for all $x$. But the issue is that i'm not completely sure why $g(U)$ is open in $\mathbb{R}^n$.

Answer 1

By definition, $g(U) = \{\phi(x) - \phi(p) \ \vert \ x \in U\} = \phi(U) - \phi(p)$. Since $\phi(U)$ is open, so is $g(U)$ (because translations trivially preserve openess).