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The Problem: As clarified in the answer to this question , a smooth function $f:[a,b] \rightarrow \mathbb{R}^n$ is an immersion at the point $p \in [a,b]$ if the derivative $\partial f(p) \neq 0$. In the context of smooth manifolds $M$ and $N$, however, one says that that a smooth function $f:M \rightarrow N$ is an immersion at the point $p \in M$ if the differential at $p$, $$f_{*,p}:T_pM \rightarrow T_{f(p)}N$$ is injective. I am trying to verify that this more general definition is equivalent to the previous one when working with curves in $\mathbb{R}^n$. According to the guidelines suggested in post on meta, I will be posting my proposed solution to this problem as an answer to the question.

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