Local character of submanifolds

Question

Let $M$ be a smooth manifold and let $A\subset M$. Then, $A$ is a submanifold of $M$ if we can find an open covering $\{A_i\}_{i\in\mathcal{I}}$ of $A$ and an open covering $\{M_i\}_{i\in\mathcal{I}}$ of $M$ such that each $A_i$ is a submanifold of $M_i$.
My attempt is this. Let $\mathcal{A}_i$ be a smooth atlas of $A_i$, which exists by hypothesis. I want to show that $\bigcup_{i\in\mathcal{I}}\mathcal{A}_i$ is a smooth atlas of $A$. The problem is to show that the atlases are compatible. Does this follows from the fact that each $A_i\subset M_i$?
The problem arises from Hircsh "Differential topology".
Thank you all in advance.

Answer 1

Maybe, all follows from the fact that all $A_i$ are submanifolds of open sets of the ambient manifold.
Let $\mathcal{M}:=\{(V_j,\varphi_j)\}_{j\in\mathcal{J}}$ be a $\mathcal{C}^r$-atlas of $M$, then we can give to each $M_i$ the structure of $\mathcal{C}^r$-differentiable manifold through the atlases $\mathcal{M}_i:=\{(V_j\cap M_i,\varphi_j|_{V_j\cap M_i})\}_{j\in\mathcal{J}j}$. Since, by hypothesis, each $A_i$ is a submanifold of $M_i$, each atlas $\mathcal{M}_i$ induces an atlas $\mathcal{A}_i$ on $A_i$, through $\mathcal{A}_i:=\{(V_j\cap A_i,\varphi_j|_{V_j\cap A_i})\}_{j\in\mathcal{J}}$. Thus, we can give $A$ the differentiable structure given by the atlas $\mathcal{A}:=\bigcup_{i\in\mathcal{I}}\mathcal{A}_i$, whose charts are compatible, since they are just the restriction of charts in a $\mathcal{C}^r$-atlas.