More seriously. Here's a link to an article about these spheroïds.

https://www.nature.com/articles/srep30614

Sometimes difficult but really interesting.

It's a little hard to see exactly how heavy the stones they were looking at are but they seem to be ranging from 0.2kg up to something around 2.2kg. I think it would be pretty difficult to sling 2.2kg and a lot of those stones are pretty heavy for slinging in my opinion but some are right around the size I would use. One thing that I think is very interesting is that the stones thrown by hand were calculated to be disabling against an impala-like animal. That's one stone being thrown by one person. They were easily killing them then with multiple people and multiple stones. Really shows how effective a stone can be and great that they were able to quantify it. A very useful little equation that we could implement for stones we know are being slung. If they did have slings and were using these stones (which to my mind is absolutely possible, I mean they were intelligent enough to specifically seek out and transport these stones from another area for a specific purpose and they were using various other tools) then there is no question that they were very effective hunters. Stones really are no joke.

Those are good points. I'm really pretty small and weak to be honest so I am one hundred percent sure they could out throw me even if I had a sling and they didn't

I think I found the olders trace of a sling.

But more seriously. Just a band of leather. This hypothesis needs a silex to cut the leather and a hunted animal. It's all.
Thank you Kilisi for your photo. 

Sorry, I realize that I used a french word in my post.
Silex is flint.
With a knife of flint, the cavemen could make a sling with the leather. It's a very easy technology.
I don't test in the true life and the archeologists don't find these artefacts.
So we don't know.
But I like this idea.