I would like to show that a torsion, coherent sheaf $\mathcal{F}$ on a regular integral curve $C$ is supported at a finite number of closed points. This is from Ravi Vakil's notes, namely part 13.7.G.
Because freeness is a stalk local condition for coherent sheaves, we know that $\mathcal{F}$ is $0$ on a dense open subset $U$.
Thus $\text{Supp}(\mathcal{F})\subset C\setminus U$, and by the assumptions we know it's supported on closed points.
My issue is showing finiteness. If I knew the curve were Noetherian this would be immediate as then the support is contained in a proper closed set.
How should I proceed from here?
Have a look at the formulation of the exercise again. He explicitly states that a regular curve is implicitly assumed to be locally noetherian and in part $b)$ (your exercise) he assumes the curve to be quasi-compact. We have
quasi-compact + locally noetherian = noetherian.
So you are done, aren't you?