I am somehow missing what are your goals - perhaps introduction to proofs, exposition to higher mathematics, or indicating what does it involve to do research?

Nevertheless, you may be interested in the book The Enjoyment of Mathematics: Selections from Mathematics for the Amateur by H. Rademacher and O. Toeplitz if you can get hold of it (very unfortunately, it is out of print, but the table of contents can be checked on amazon). It contains short pieces not requiring anything beyond high school math, yet many of them are full of elegance and joyful to read. As the students certainly should be able to get through the chapters independently, covering few of them can provide a good start.

In a similar direction, but more advanced, is Proofs from the book.