But proving brain activity is not easy either is it?
It is very simple - aside from machines to measure brain activity, that you are able to read any post here indicates brain activities, since more than one process is 'running' as you do so. A dead person has no brain activity going on and, therefore, cannot read posts in the Forum.
Next time you hear sound, feel hungry, dream, experience pain, remember something from your childhood etc etc etc - your brain is active: it is even active in ways you are normally unaware of, such as controlling your respiration (you don't have to remember to keep breathing).