1. The completeness property of R.

2. The Archimedean principle in R.

3. Limit of a sequence.

4. Convergent sequences.

5. Monotone and bounded sequences.

6. Cauchy sequences.

7. Subsequences and limit points. Liminf, and limsup.

8. Bolzano-Weierstrass Theorem.

9. Open sets, closed sets, bounded sets and compact sets in R.

10. Limits of real valued functions.

11. Definition of limits by neighborhoods.

12. Definition of limits by sequences.

13. Continuous functions on R.

14. Sequence definition and neighborhood definition of continuity.

15. Boundedness of continous functions on compact intervals.

16. The extreme value theorem.

17. The intermediate value theorem.

18. Uniformly continuous functions.

19. The sequential criterion for uniform continuity.

20. The derivative of functions.

21. Roles Theorem.

22. Mean value theorem.

23. Generalized Mean value theorem.

24. Taylor Theorem with remainder.

25. Lâ€™Hospital,s rule.