This is problem 20.1 from Loring Tu's An Introduction to Manifolds:
Let $I$ be an open interval, $M$ a manifold, and {$X_t$} a 1-parameter family of vector fields on $M$ defined for all $t \neq t_0 \in I$. Show that the definition of $\lim_{t \rightarrow t_0}X_t$ in (20.1), if the limit exists, is independent of coordinate charts.
The definition in question is $\lim_{t \rightarrow t_0}X_t|_p = \sum_{i=1}^{n}\lim_{t \rightarrow t_0}a^i(t,p)\frac{\partial}{\partial{x^i}}|_p$, where $(U, x^1, ..., x^n)$ is a coordinate chart about $p \in M$.
Here's what I've attempted so far: Let $p \in M$, and let $(U, \phi)=(U,x^1,...,x^n), (V, \psi)=(V,y^1,...,y^n)$ be coordinate charts about $p$. Then $X_t|_p = \sum_{i=1}^{n}a^i(t,p)\frac{\partial}{\partial{x^i}}|_p = \sum_{j=1}^{n}b^j(t,p)\frac{\partial}{\partial{y^j}}|_p$. Applying both sides to $y^k$, we get $\lim_{t \rightarrow t_0}b^k(t,p)=\sum_{i=1}^{n}\lim_{t \rightarrow t_0}a^i(t,p)\frac{\partial{y^k}}{\partial{x^i}}|_p=\sum_{i=1}^{n}\lim_{t \rightarrow t_0}a^i(t,p)\frac{\partial{(\psi \circ \phi^{-1})^k}}{\partial{r^i}}|_{\phi(p)}$, where $r^i$ are the standard coordinates.
At this point I'm sort of lost. I think there's probably something pretty obvious that I'm missing but I haven't noticed what it is. Any tips?
