Consider the following parameterization of a sphere of radius $R>0$ given as $\Gamma:[0,2\pi]\times [0,\pi]\to\mathbb{R}^3$ given by $(\theta,\phi)\mapsto (R\cos\theta\sin\phi,R\sin\theta\sin\phi, R\cos\phi)$. Now let $0<r<R$ and consider the upper part of the sphere $C=\{(\theta,\phi):0\leq\theta\leq 2\pi, 0\leq\phi\leq r/R\}$.
Show that the distance between $P=(0,0,1)$ and $\Gamma(Q)$ with $Q\in C$ is at most $r$.
Now my idea was to find a curve $\alpha(t)$ that connects $P$ and $\Gamma(Q)$ and then finding an upper bound for the integral $$\int_a^b ||\Gamma(\alpha(t))'||dt$$ But I can't seem to find a relatively easy way to find such a curve or even still to calculate such integral. Maybe I'm looking at the wrong approach?
Any help is greatly appreciated.
To show that the meridians are length-minimizing, note that the first fundamental form of the sphere in spherical coordinates (such as the ones you chose) is diagonal. Therefore when calculating length, the integrand will be $$\sqrt{g_{11} ({\alpha_1}')^2 +g_{22}({\alpha_2}')^2},$$ where $g_{11}$ corresponds to your coordinate $\theta$ and $g_{22}$ corresponds to your coordinate $\phi$. But $\sqrt{g_{11} ({\alpha_1}')^2 +g_{22}({\alpha_2}')^2}\geq \sqrt{g_{22} ({\alpha_2}')^2}$ from which the result follows.