I am trying to prove the lemma from the Lee, Introduction to Smooth Manifolds book:
Lemma 3.16. Let $M$ be a smooth manifold. If $p \in M$ and $X \in T_pM$, there is a smooth vector field $\tilde X$ on $M$ such that $\tilde X_p=X$.
The proof in the book uses a bump function to construct the vector field $\tilde X$, however I made a proof without a bump function which is therefore suspiciously simpler and would like to verify if it is correct, otherwise what is wrong:
Let $X=X^i \frac{\partial}{\partial x^i}|_p$, then define the vector field $\tilde X:M \to TM$ by $r \mapsto X^i \frac{\partial}{\partial y^i}|_r$ where $r \in V$ for some chart $(V,\psi) \in \mathcal{A}_M$ with $\psi(V)$ having the local coordinates $y^i$. Then $\tilde X$ is a vector field since given the projection $\pi :TM \to M$ it is true that $\tilde X \circ \pi = id_M$. $\tilde X$ is smooth iff for any smooth charts $(U, \phi), (V, \psi) \in \mathcal{A}_M$ in an atlas for $M$ with the correspondent smooth chart $(\pi^{-1}(V), \Psi:\pi^{-1}(V) \to \mathbb{R}^{2n}) \in A_{TM}$ the function $\Psi \circ \tilde X \circ \phi^{-1}$ is smooth where $\Psi:v^i \frac{\partial}{\partial y^i}|_r \mapsto (y^1(r), ..., y^n(r), v^1, ..., v^n)$ with $y^i$ being a local coordinate of $\psi(V)$.
From the definition $\psi \circ \tilde X \circ \phi^{-1}:x \mapsto (y^1(\phi^{-1}(x)), ..., y^n(\phi^{-1}(x)), X^1, ..., X^n)$ where the $y^i$ are the local coordinates, i.e. $\psi(r)=(y^1(r), ..., y^n(r))$. Since the transition function $\psi \circ \phi^{-1}$ is smooth by the compatibility of the charts, all components of $\psi \circ \tilde X \circ \phi^{-1}$ are smooth, therefore $\psi \circ \tilde X \circ \phi^{-1}$ is smooth as required.