The Cauchy–Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities is a graded problem book that will teach them the principles and practice of mathematical proofs like no other book. Here is its MAA review: [0]. A pirate PDF is a Google search away. Take a look and see if you like it.

[0] https://www.maa.org/press/maa-reviews/the-cauchy-schwarz-mas...