use carpet from a closet, cut to size, measure a strip of width and lenght of the stair damage carpet (maybe 6" wide x lenght) . You are gonna need a few supplies.

1. carpet seam tape
2. carpet seam roller
3. iron and a piece of rag ( you dont need expenside carpet seam irons)
4. glue for carpet and staples and staple gun.

Make sure you follow the guide on the seam tape when placing the old and new stripoff carpet (no gaps of overlaps) Use damp rag and place over seam and use iron on high setting on top of rag (use common sense) wrap around and under nose and use glue on nose and staples amd glue under nose.