I am trying to solve this question:
Show that the Cartesian product $f_1 \times f_2: N_1 \times N_2 \to M_1 \times M_2,$ of two embeddings $f_1, f_2$ is again an embedding.
My definition of an embedding is:
My definition of an $n$-dimensional smooth submanifold is:
It is easy for me to prove that the composition of two diffeomorphisms is again a diffeomorphism. But how can I prove that the composition of two submanifolds is again a submanifold, it is for me true according to intuition.
The hint at the back of the book is:
My question is:
I do not know why the hint is saying reduce to the case where $f$ and $g$ are inclusions of submanifolds, how will this help us in the proof? could someone help me in answering this question please?
Also, what does it mean to "rearrange some coordinates" that is mentioned in the hint? could someone help me answer this please?
Some facts:
Assume that $f_1: N_1 \to M_1$ is an embedding, then $f(N_1) \subset M_1$ is a differentiable submanifold and $f_1: N_1 \to f(N_1)$ is a diffeomorphism.
Also, Assume that $f_2: N_2 \to M_2$ is an embedding, then $f(N_2) \subset M_2$ is a differentiable submanifold and $f_2: N_2 \to f(N_2)$ is a diffeomorphism.