You're missing the point. You can't prove your existence if you have to assume you exist first; that just doesn't make sense as an argument. You can say that existence and 'doing things' go hand in hand and can't be separated, if you really want to, but you can't then prove one of them simply by inferring the other one. If something's existence is in doubt then so is whether it is 'doing things', because 'doing things' can only happen if the thing exists, which hasn't been proven yet.