Sure, but it's impossible to prove that brain training has no benefits, all we can say is that essentially every well-run study has failed to find any benefits. They're just saying that stuff to cover their bases - most neuroscience researchers don't believe that "brain training" is anything more than just wishful thinking.

It is impossible to prove, and yet the title claims it to be true.

Given the evidence, it is a reasonable conclusion. Very little is ever "proved". Rather, we make conclusions based on the available evidence. If the evidence changes, we revisit our conclusions.

Whoever downvoted you, has probably never heard of the axiom: "Nothing can ever be proven true, because that means you would have to consider every possibility. The best that we can hope for is to prove something is not true by finding exceptions. After many failures at finding exceptions, we just state that some things are true until an exception is found."

The thing is, researchers commonly put in throwaway lines like your latter quote to avoid themselves from being embarrassed by some new study that comes out in the future. The key to comprehension is appreciation which portions are fundamental to the message and which parts are throwaway.

I thought it was to ensure on-going funding. If you don't conclude that further research is necessary then that's surely the worst possible thing for your continuing to be paid to do said research.