I'm new to manifolds and in exercise 4.2.1 from J.J. Duistermaat's Multidimensional Real Analysis I, I have to prove that $V = \{x\in\mathbb{R}^2 : x_2=x_1^3\}$ is a $C^{\infty}$ manifold.
The definition given in the book is
First of all I do not know how to go from proving this for a single $x$ to going to the global case. For a single $x$, say $x=(1,1)$ I did the following:
Let $U=\{(x_1,x_2)\in\mathbb{R}^2\mid 0<x_1<2,0<x_2<8\}$ be an open neighbourhood of $x$. Let $W=\{x_1\in\mathbb{R}\mid 0<x_1<2\}$ be an open subset of $\mathbb{R}^1$. (so $d=1$ per the definition). Define $f(x_1)=x_1^3$. This is a $C^{\infty}$ mapping.
Then $V\cap U=\{(x_1,x_2)\in\mathbb{R}^2\mid 0<x_1<2, 0<x_2<8, x_2=x_1^3\}=\{(w, f(w))\in\mathbb{R}^2\mid w\in W\}$.
Therefore $V$ is a $C^{\infty}$ submanifold of $\mathbb{R}^2$ at $x$ of dimension $1$. (Quite a laborious proof)
How would we prove the global case? I see this proof can be mimicked for any chosen $x$, but how to prove it?
The graph of a $C^k$-map $f : W \to \mathbb R^{n-d}$ defined on an open $W \subset \mathbb R^d$ is always a $C^k$-submanifold of $\mathbb R^n $ of dimension $d$. We have $$\text{Graph}(f) = \{(w,f(w)) : w \in \mathbb R^d \}.$$ For each $x \in \text{Graph}(f)$, the whole space $\mathbb R^n$ is an open neighborhood of $x$ in $\mathbb R^n$ such that $\text{Graph}(f) \cap \mathbb R^n = \text{Graph}(f)$ which proves our claim.
Now let $f : \mathbb R \to \mathbb R, f(w) = w^3$. This is a $C^\infty$-map such that $V = \text{Graph}(f)$.