Let $S \subset M$ be an immersed submanifold and $i:S \to M$ the inclusion map. In this case, why is the differential $i_*: T_p S \to T_p M$ injective, i.e. $i$ is an immersion?
The definition of an immersed submanifold means that $S$ is an image of some $f:N \to M$ that is a one-to-one immersion. How does this imply that the inclusion map from $S$ to $M$ is an immersion?
Let $\tilde{f}:N\to S$ be the bijection obtained by restricting the codomain of $f$. As the topology and the differentiable structure is induced by $\tilde{f}$, $\tilde f$ is a diffeomorphism. Then $i=f\circ\tilde f ^{-1}$ is differentiable. By the chain rule
$$D_pf=D_p(i\circ\tilde f)=D_{\tilde f(p)}i\circ D_p\tilde f. $$
As $f$ is an immersion $D_pf$ is injective. Also $D_p\tilde f$ is invertible so $D_{\tilde f(p)}i$ must be injective for all $p$ and hence $i$ is an immersion.