Let $\alpha ,\beta :[a,b] \to \Bbb R^n$ be two paths from $0$ to $x$ and $$\beta(t) = \frac{x.\alpha(t)}{x.x}x$$. To show that length of $\beta$ is bounded above by length of $\alpha$.
length of $\beta$ = $$\int_a^b |\beta'(t)| dt = \int_a^b |\frac{x.\alpha'(t)}{x.x}x| dt \leq \int_a^b \frac{1}{|x|^2} |x||\alpha'(t)||x| dt = \int_a^b |\alpha'(t)| dt$$
Is the method correct?