ScalingStacks

Proof. Without loss of generality, we can assume $i_{1}<i_{2}$. The lemma follows by applying the intermediate value theorem to $\gamma_{2}(t)-\gamma_{1}(t)$ and using Lemma 6.2.2, considering four cases according to the signs of $j_{2}-j_{1}$ and $\sigma(i_{2})-\sigma(i_{1})$. ∎