$\newcommand{\M}{\mathcal{M}}$
$\newcommand{\N}{\mathcal{N}}$
$\newcommand{\R}{\mathbb{R}}$
$\newcommand{\Vol}{\operatorname{Vol}}$
$\newcommand{\Det}{\operatorname{Det}}$
$\newcommand{\Volm}{\operatorname{Vol}\_{\M}}$
$\newcommand{\Voln}{\operatorname{Vol}\_{\N}}$

This is a self-answered question, which I put here since it wasn't obvious for me. (Tangent bundle stuff can be confusing.)

Let $\N$ be a smooth **compact** $n$-dimensional Riemannian manifold, and let $i:\N \to \R^D$ be an isometric embedding. Let $q\_k,q \in \N$, $w\_k \in T\_{q\_k}\N,w \in T\_q\N$.

> 
> *Claim:* Suppose that $di\_{q\_k}(w\_k) \to di\_{q}(w)$. (this is a convergence of a sequence of vectors in $\R^D$). Then $w\_k \to w$ w.r.t the topology of $T\N$.
> 
> 
> 

Of course, I would be happy to see other approaches.