Here is a very interesting course in Real Analysis which was held by professor Francis Edward Su at Harvey Mudd College, USA. I think that students (especially Romanian students) will find this course really nice and more focused on the interaction between student-professor.

**REAL ANALYSIS COURSE**

Lecture 1 (Constructing the Rational Numbers)

Lecture 2 (Properties of )

Lecture 3 (Construction of the reals)

Lecture 4 (The Least Upper Bound Property)

Lecture 5 (Complex Numbers)

Lecture 6 (The Principle of Induction)

Lecture 7 (Countable and Uncountable Sets)

Lecture 8 (Cantor Diagonalization and Metric Spaces)

Lecture 9 (Limit Points)

Lecture 10 (The Relationship Between Open and Closed Sets)

Lecture 11 (Compact Sets)

Lecture 12 (Relationship of compact sets to closed sets)

Lecture 13 (Compactness and the Heine-Borel Theorem)

Lecture 14 (Connected Sets, Cantor Sets)

Lecture 15 (Convergence of sequences)

Lecture 16 (Subsequences, Cauchy Sequences)

Lecture 17 (Complete Spaces)

Lecture 18 (Series)

Lecture 19 (Series Convergence Tests, Absolute Convergence)

Lecture 20 (Limits and Continuity)

Lecture 21 (Continuous Functions)

Lecture 22 (Uniform Continuity)

Lecture 23 (Discontinuous Functions)

Lecture 24 (The Derivative and the Mean Value Theorem)

Lecture 25 (Taylor’s Theorem, Sequence of Functions)

Lecture 26 (Ordinal Numbers and Transfinite Induction)