Rewrite
1. It occured to me that you might like to come round this evening. (WONDERING)
2. Our arguments over politics go back year. (ARGUING)
3. My son has finally come to accept that there's no such thing as a free lunch. (DAWNING)
4. Who is the organiser of this event? (RUNNING)
5. I think he's at last beginning to agree with me? (ROUND)