In the book the definition of a a vector field over $U$(open)$ \subseteq S^n$ is given by a continuous map $s: U \to T(U)$ such that $p_U \circ s=id_U$ where $p_U$ is the base point projection from $T(U)$ to $U$.
Now given $v_0\in S^n$ show that there is an open neighbourhood $U \subset S^n$ of $v_0$ and vector fields $s_i:U \to T(U)$, $1 \leq i \leq n$ such that $\{s_1(v),s_2(v),...,s_n(v)\}$ is a basis of the vector space $T_v(S^n)$, $\forall v \in U$.
I approached to extend the basis $v_0$ to an orthonormal basis using Grahm Schmidt normalization process where at each step we have to divide by scalar and subtract all of which are continuous. So I thought this is continuous. Is there any other way to prove continuity using the inverse image of an open set is open?