Well you can't prove anything using the current scientific limitations, so you'll just have to go with the fact that everything that we know about the world is an indirect assumption. (No, I'm serious. Almost every working theory in chemistry and physics are just assumptions that ended up working.)
And, in what way is it not convincing? Or rather, how else can you explain it, without admitting that they have at least some arithmetic capability?
Between the fact that it's a documentary or the fact that I still don't see a paper to read? Interest in object =/= computational ability. Yes, it MAY signify perception of some sort but that is in no way concrete evidence for a working theory. The cool thing about theories is that they repeatedly work and the funny thing about coincidences is that they don't need to be proved.
So after actually finding the paper and reading it, things make a little more sense. In one of the many actual studies they performed, they monitored brain activity with electrodes and compared the correct solution signals in the brain with the incorrect brain response THEN compared these responses to that of adults. However, they only monitored the 15 of the 24 babies with the 15 being the babies that actually detected the error so there's some slight lapse in data there. Full paper link below if you can access it.
http://www.pnas.org/content/103/33/12649.full
Well you can't prove anything using the current scientific limitations
http://en.wikipedia.org/wiki/Proof_(truth)