I attempt to understand the proof of the following lemma from An Introduction to Manifolds by Loring Tu (Second Edition, page no. 51).
Lemma 5.8. Let $\lbrace (U_{\alpha}, \phi_{\alpha}) \rbrace$ be an atlas on a locally Euclidean space. If two charts $(V, \psi)$ and $(W, \sigma)$ are both compatible with the atlas $\lbrace (U_{\alpha}, \phi_{\alpha}) \rbrace$, then they are compatible with each other.
My Question: I understand that $\sigma \circ \psi^{-1}$ is $C^{\infty}$ on $\psi(V \cap W \cap U_{\alpha})$. But I don't understand how $p$ being an arbitrary point of $V \cap W$ implies that $\sigma \circ \psi^{-1}$ is $C^{\infty}$ on $\psi(V \cap W)$ on the second last line.
An explanation would be really helpful.
But, Tu has shown for an arbitrary $p\in V\cap W$, that $\sigma\circ\psi^{-1}$ is smooth at $\psi(p)$. Clearly the claim follows. Remember, for each $p$ there's a $U_\alpha$. So, $\psi(V\cap W)=\bigcup_{p\in U\cap V}\psi(V\cap W\cap U_\alpha)$. Now if a function is smooth on a family of open sets, it's smooth on their union.