When facing Peeves in Restricted section of the library, use a full force Flipendo spell on him first to make him hide in armor suits. When he hides, immediately use a full force Flipendo spell on the armor suit where he hides. It will be destroyed. Be careful, as it sometimes gives off Dungbombs which costs 3 points of life. Repeat the steps, as there are four armor suits. When all are destroyed, use a full force Flipendo spell again on Peeves until he gives up.