What would a caveman have done?
I suppose the basic answer to that one is, generally, not have lived in a cave: but as I'm sure* I've seen chimps fanning themselves with leaves** then we can assume, I think, that fans have been around for a very long time. In fact, they must have been around longer than there have been words to describe them.
Our word fan, though, started off describing the big basket-like thing you use for throwing ears of wheat skywards so the wind can blow away all the leafy husks. Fans only started being used for fluttering and flirting in the mid 1500s.
Which is, admittedly, still quite a long time ago.
A fan is also, of course, an enthusiast for something or other. I'm a fan of words, Baroque music, roses, frogs, Gershwin, fashion, and loads of other things that haven't occurred to me at the moment.
This probably makes me weird, but certainly happy.
Spot the frippet: fan. Apart from the obvious sorts of fans (look out for the football shirts!) you might be lucky enough to be able to find a fan-tailed bird or a fan-leafed plant.
Fan. The word for the wavery thing comes the Old English fann, and before that from the Latin vannus. The enthusiast sort of fan is short for fanatic, which is from the Latin fānāticus, which means belonging to a temple, and so inspired by God into a frenzy. Church must have been much livelier in those days.
*I use the expression I'm sure in the usual way to mean I have some vague recollection.
**Though there is just a slight possibility that was a character in a Disney film.