In Lee's Introduction to Smooth Manifolds there is a exercise to proof that if $f: M \to \mathbb{R}$ and $g: M \to \mathbb{R}$ are smooth, so is $fg$. The question I'm having is the following:
By definition $f: M \to \mathbb{R}$ is smooth in $p \in M$ if there exists a smooth chart ($U, \varphi$) such that $U$ is a neighbourhood of $p$ and $f \circ \varphi^{-1}$ is smooth. Then because $g$ is smooth, there exists a smooth chart ($V, \psi$) such that $V$ is a neighbourhood of $p$ and $g \circ \psi^{-1}$ is smooth. From here I should deduce that $fg$ is smooth. However I'm wondering how can I be sure that there exists a smooth chart ($W, \theta$) such that $fg \circ \theta^{-1}$ is smooth. How does this work if $\varphi^{-1}$ and $\psi^{-1}$ map $p$ to totally different areas in $\mathbb{R^n}$ or if all compatible charts with $f$ are not compatible with $g$? Can there even be such cases?
The definition of Differentiable function on smooth manifold, does not depend on the choice of local chart. Since if $f:M\to \mathbb{R}$ be Differentiable at $p\in M$ and $(U,\varphi)$ be the corresponding local chart. For each local chart $(V,\psi)$ where $p\in V$, we have the map $\varphi \circ \psi^{-1}:\psi(W)\to \varphi (W)$ is $C^{\infty}$, where $p\in W=U\cap V$. Notice that $(W,\psi)$ is a smooth chart due to its apparent smooth compatibility with other smooth charts and the maximality of the determined atlas.
Observe that $f\circ \psi^{-1}|_{\psi(W)}=f\circ\varphi^{-1}\circ\varphi \circ \psi^{-1}|_{\psi(W)}$. Because $M$ is a smooth manifold, $\varphi \circ\psi^{-1}|_{\psi(W)}$ is $C^{\infty}$ and $f \circ \psi^{-1}$ is $C^{\infty}$ too. Hence, their composition is $C^{\infty}$.
Now, since $\cdot$ and $+$ are smooth, so do $(f\cdot g)\circ \psi^{-1}|_{\psi(W)}=(f \circ \psi^{-1}|_{\psi(W)})\cdot(g\circ \psi^{-1}|_{\psi(W)})$ and $(f+ g)\circ \psi^{-1}|_{\psi(W)}$.