The heat "sealing" method to prevent fraying proved to be inadequate.
So I bought a product from Singer called "Fray No More" at Wal-Mart for $4.00. (The only way to buy it was in a twin-pack with another product called "Sew No More," which apparently is some kind of adhesive thread replacement. I have no desire to try it.)
It's a little bottle of clear liquid with a pointed applicator tip, like a bottle of Elmer's glue. I fortunately had saved the scrap pieces of bands, so I tried it on one first to see if it would stain. It didn't, so I forged ahead with all five bands. Just release the clear gel onto any cut edges and let dry for about 30 minutes. So far, so good.