= Solution
Suppose that two closed geodesics $\gamma_1$ and $\gamma_2$ were disjoint. Under the usual convention that a closed geodesic curve is simple, the <Jordan curve theorem> on the sphere shows that they bound an annulus $A\subset S$. Its Euler characteristic is zero. Give $A$ the induced orientation and apply the <Gauss-Bonnet theorem>:
$$
\int_AK\,dA+\int_{\partial A}\kappa_g\,ds=2\pi\chi(A)=0.
$$
Both boundary components are geodesics, so their <geodesic curvature> vanishes. It follows that
$$
\int_AK\,dA=0.
$$
This is impossible because $K>0$ everywhere and $A$ has positive area. Hence the two closed geodesics must intersect, as summarized by the <intersection of closed geodesics on a positively curved sphere>.
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