Existence of Local Coordinates

Theorem
Let $X^n$ be an $n$-dimensional manifold.

Let $U$ be a neighborhood of a point $p \in X^n$.

Then there exist local coordinates on $U$.

Proof
There is a homomorphism $\phi: U \to \R^n$, since $X^n$ is a manifold.

Let $y_i$ be coordinates in $\R^n$, so that any point $z \in \R^n$ has a unique representation $y_1, \dots, y_n$.

Then define $x_i: X^n \to \R$ as $x_i = y_i \circ \phi$.