Let $M \subset R^n$ be a manifold and $x \in M$ a point on it. I want to prove that if $h \in T_xM$ (the tangent space at x) then for every $\epsilon$: $\text{distance}(x+\epsilon h) = o(\epsilon)$, where $\epsilon \to 0$.

So I started like this:

Since $h \in T_xM$, then there exists $a>o$ and a smooth function $\gamma:(-a,a) \to M$ such that $\gamma(0) = x$ and $\gamma'(0) = h$.

Now I need to show that $$\lim \limits_{\epsilon \to 0} \frac{distance(x+\epsilon v, M)}{\epsilon} = 0$$

However, I am not really sure what distance means. If it is the euclidean distance, then I would choose some $y \in M$ and try so prove: $$\lim \limits_{\epsilon \to 0} \frac{||x+ \epsilon v -y||}{\epsilon} = 0$$

but I am not sure how to do it, since there is no guarantee that $y \in Im(\gamma)$ and thus I can't use $\gamma$/

Any help would be appreciated


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