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After a fixed orthogonal change of ambient coordinates, a neighbourhood of $p$ in $X$ is the graphof 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 formThe convergence of $w_i$ implies convergence of its first component, say $v_i\to v$. Since $u_i\to u$ and $Dg$ is continuous,This proves the closedness of tangent spaces under convergent base points.

Conversely, suppose $d>0$ and write $w=(v,Dg(u)v)\in T_{p}X$. Choose a nonzero $e\in {\mathbb{R}}^d$ and positive $t_i\to 0$ with $u_i=u+t_{i}e\in U$. Thenand $p_i\to p$, $w_i\to w$. Every tangent vector arises as such a limit from distinct base points.