Book for real analysis:
The syllabus that needs to be covered is
Elementary set theory, finite, countable and uncountable sets, Real number system as a complete ordered field, Archimedean property, supremum, infimum. Sequences and series, convergence, limsup, liminf.
Bolzano Weierstrass theorem, Heine Borel theorem.
Continuity, uniform continuity, differentiability, mean value theorem.
Sequences and series of functions, uniform convergence.
Riemann sums and Riemann integral, Improper Integrals.
Monotonic functions, types of discontinuity, functions of bounded variation
I checked out the obvious recommendations given in the answers here which mainly include Rudin,Apostol,Pugh.
But the problem is none of them don't contain enough examples and material to learn the above topics on my own.
Having problem sets is a must have for the books.
I need a book that contains enough examples and good exercises(with hints possibly) that can help me learn the material and also help to prepare for the entrance exams that ask questions on real analysis.
Any help will be much useful.