Real Analysis (M1101)

For first-year students at the Lebanese University, Faculty of Sciences MISPE Students . M1101 takes the calculus you know from school and makes it rigorous. Instead of just computing limits and derivatives, you'll learn why they work: precise definitions, epsilon-delta reasoning, and proofs of the major theorems. This course walks you through the entire syllabus step by step, so you understand the theory, master the standard techniques, and can handle the exercises you'll face in exams. How the course works Recorded lessons: Every chapter is explained in clear, structured videos. You can watch at your own pace, pause, and rewatch before exams. Live sessions: Regular live classes where we solve exercises together, go over common exam questions, and answer your questions directly. Exercise-focused: We work through chapter exercises and exam-style problems, not just the theory. Partial and Final Past Exams Solution . What you'll learn Real Numbers: The structure of ℝ, upper and lower bounds, supremum and infimum, and the completeness property. Numerical Sequences: Limits and convergence, monotone and bounded sequences, adjacent sequences, subsequences, and Cauchy sequences. Limits and Continuity: Limits of functions on ℝ, continuity, and the key theorems on continuous functions (intermediate value theorem, extreme values). Differentiation: Derivatives and their properties, Rolle's theorem, the Mean Value Theorem, and Taylor's formula. Elementary Functions: Trigonometric, logarithmic, exponential, and hyperbolic functions (sin, ln, exp, sinh, and more) along with their inverse functions. Finite Expansions: Taylor expansions, approximating functions, and using expansions to compute difficult limits. By the end of the course, you will be able to: Write rigorous proofs using precise definitions of limits and continuity Study the convergence of sequences and the behavior of functions Apply the Mean Value Theorem and Taylor's formula Use finite expansions to compute limits and approximations quickly Who it's for This course is for first-year Faculty of Sciences MISPE students taking M1101 at First Semester .