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How to do it has been sorta laid out. I would prefer to do it in writing and then following up with a phone call a week or so later.
Explain what the sport is, what you are looking for, & what you can do for them. This is a way for them to advertise and you need to sell this as a viable option.
Yeah, it is good if your sponsor can tell down the road that they got business out of their investment. Not everyone who goes to that business is going to thank them for being a sponsor so having some way for them to know that those in attendance took notice of their support and in return are supporting them.
I would lay out a few options (depending on the business).
Gold: Cash & Product
No Bronze on purpose
So, for example you get the local pizza place to be a sponsor. They provide lunch for free to the players. The players would think that is great and then perhaps in the player pack a special $2 coupon (or whatever) is placed in the player packs. That way when the players then later order a pizza, the owner can see a return from the disc golfers. So, Joe's Pizza Shack is a Silver Level Sponsor and hopefully one that can be counted on in the future.