The simple way to kill bedbugs is by wrapping the mattress with cover that is closely woven so no baby bugs can get through it. E.g.: Plastic, Rubber, etc. Choose the light color so you could spot the bedbugs easily and spray them with insect spray like Raid, Baygon, etc.
After that, you could cover the mattress with a bed sheets you like. You must wash the sheets every week and make sure there is no bug there.