= Solution
After a fixed orthogonal change of ambient coordinates, a neighbourhood of $p$ in $X$ is the graph
$$
\{(u,g(u)):u\in U\subseteq\mathbb R^d\}
$$
of a smooth map $g:U\to\mathbb R^{n-d}$. Write $p_i=(u_i,g(u_i))$ for all sufficiently large $i$. Every tangent vector has the unique form
$$
w_i=(v_i,Dg(u_i)v_i).
$$
The convergence of $w_i$ implies convergence of its first component, say $v_i\to v$. Since $u_i\to u$ and $Dg$ is continuous,
$$
w_i\longrightarrow(v,Dg(u)v)\in T_pX.
$$
This proves the <closedness of tangent spaces under convergent base points>.
Conversely, suppose $d>0$ and write $w=(v,Dg(u)v)\in T_pX$. Choose a nonzero $e\in\mathbb R^d$ and positive $t_i\to0$ with $u_i=u+t_ie\in U$. Then
$$
p_i=(u_i,g(u_i))\ne p,
\qquad
w_i=(v,Dg(u_i)v)\in T_{p_i}X,
$$
and $p_i\to p$, $w_i\to w$. \b[Every tangent vector arises as such a limit from distinct base points.]
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