You have used the work correctly in the post to which I am replying. I have fanned myself and others with many pieces of cardboard over the years, lol. You can also write:
My son told me to fan him with a piece of cardboard.
In this thread, you are asking about how the word 'fan' can be used as a verb. Usually, it is a noun.
We can have some fun with English if we compare that example with a similar example, which can be found when using the word bludgeon. Maybe it will be interesting for people to Google around about the way to use 'bludgeon' as a noun or as a verb.
Lol, it is possible for your son to fan himself with a fan, and it is likewise possible for me to bludgeon myself with a bludgeon. That is a funny example.
Student or Learner