Why are these books from Dover Publications so cheap?
Because the copyright has expired, or the copyright owner has put the work in the public domain due to bankruptcy, or similar.
I used this for my undergrad analysis class and then relied on it as a foundation for reading Rudin/Royden in grad school. Killed qualifying exams. Use it. Sure, it's all Riemann integration but you will not suffer until you get to measure-theoretic integration later.
We used it at the University of Texas at Austin for the first semester in Real Analysis. I found it very clear and easy to follow.