I was reading the paper by Masur-Minsky (Geometry of the Complex of Curves I: Hyperbolicity) where they show the curve complex $ C(S)$ to be $ \delta$ -hyperbolic. There given a surface $ S$ with conformal structure $ x$ , they define the extremal length of $ Ext_{x}(\alpha)$ of a curve $ \alpha$ to be the reciprocal of the modulus of the largest annulus conformal embedded into $ S$ .

I wanted to understand a statement they give in Lemma 2.4, in the title and repeated below here:

*“There exists $ e_0(S)$ such that the shortest nonperipheral curve on $ (S, x)$ has extremal length at most $ e_0$ .”*

I worry that this likely obvious. They make a comment that hyperbolic length could be used as well and I’ve heard of upper bounds for the lengths of systoles on closed surfaces. I think I could use this as one way to believe the result but that might be overkill.