to hold the carpet with the hard scuffing that the tread nose takes with the walking up and down.
I would consider using a heavy duty staple gun to tack the carpet to the underside of the nose and then again at the back of the tread next to the riser. I would make sure that my staples are not spaced but next to each other all along the nose area. There are special carpet staples that have a flat area that prohibits the staples from tearing out the area they are holding. Also get stainless steel incase the carpet gets wet and you DO NOT want rust marks.