Closedness of tangent spaces under convergent base points
= Closedness of tangent spaces under convergent base points
For an embedded smooth manifold $M\subseteq\mathbb R^n$, if $p_i\to p$, $w_i\in T_{p_i}M$, and $w_i\to w$, then $w\in T_pM$. In a local graph representation $M=\{(u,g(u))\}$, write $w_i=(v_i,Dg(u_i)v_i)$ and use continuity of $Dg$.