Armbands and suits.

[Packin' Heat]

In a number of films I've seen that take place in the Golden Era, particularly the 1930's, I notice some people wearing armbands on their shirts (on both sleeves every time I've seen it). Below is an example from The Sting:

Can anyone tell me what they're called or what they're for?

 

[Dinerman]

http://www.thefedoralounge.com/showthread.php?6345-Shirt-Sleeve-Garters/page7