MY ATTEMPT: Let $S^2=\{(x, y, z)\in \mathbb{R}^3| x^2+y^2+z^2=1\}$ be a compact surface. Defining for every $n\in \mathbb{N}$ a regular parametrized curve $\alpha_n:\mathbb{R}\rightarrow\mathbb{R}^3$ given by $\left(0, \frac{1}{n}, t \right)$, such that $t\in (-2, 2)$ (for example). Then the union $C=\displaystyle\bigcup_{n\in \mathbb{N}}\alpha_n$ define a simple curve that cuts $S^2$ transversaly into a countable (infinity) quantity of points.
MY DOUBT: How can I extend this proof to connected curves?