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

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.