I'm 99 percent sure the fan lost a belt; please confirm

My new desktop build's CPU didn't stop working, but the system was saying "fan error" at boot, so I looked in the case. First I noticed that yes, it had stopped; then I noticed that below it, a black rounded rubber band was now hanging loosely on a corner of the heatsink. Well, that pretty much tells me what must have happened: the rubber band must be a belt (what else? I didn't have any rubber bands in the case) and I need to buy a new fan. I know it's obviously broken, but am I right? A belt popped loose? No wonder I heard the fan clank before it started.

I immediately shut down and went back to this laptop. Oh, well, that was an old fan left over from my last desktop computer (but used for only a year or two). I'm off to get a new one.