In proving one result in Euclidean space $\mathbb{R}^n$, I need the following lemma:
Let $p$ be a point in a open ball $B$ and let $q$ be a point outside of the closed ball $\overline{B}$. For any smooth curve $\gamma: [a,b]\to\mathbb{R}^n$ connecting $p$ and $q$, there exists at least one and also only finitely many intersections between $\gamma([a,b])$ and $\partial B$.
Is this lemma true? This looks intuitively correct, nevertheless, how can one rigorously prove this?