I am studying differential geometry and in the lecture notes I came across this lemma. For the proof it is said to be a good exercise, but I have my doubts about the prove of my part and no clue how to do the other way around.
Given is an embedded submanifold $M \subset N$ and another manifold $P$. I want to prove that $f:P \to M$ is smooth iff $f:P\to N$ is smooth.
My attempt: If $f:P \to M$ smooth then $f:P \to N$ smooth. Let $\phi$ be a chart of $P$ and $\chi$ a chart of $M$ then we know: $\chi\circ f\circ\phi^{-1}$ is smooth. Let $i$ be the inclusion map from M to N then: $\chi \circ (i\circ\ f)\circ \phi^{-1}$ is also smooth.
I don't think this is the right approach any help or tips would be appreciated.