Foundations of Real Analysis for MSc Entrance Exams

Mastering real analysis is the single biggest hurdle for students aiming to clear competitive MSc mathematics entrance exams. This course demystifies rigorous proofs and abstract concepts, transforming them into intuitive, logical steps. Through clear written explanations, step-by-step proof breakdowns, and targeted practice problems, you will transition from memorizing formulas to deeply understanding mathematical analysis. You will develop the analytical thinking needed to tackle challenging exam questions with confidence. What you'll learn: Understand foundational concepts of real numbers, bounds, and point-set topology; Analyze the convergence of sequences and infinite series using rigorous convergence tests; Apply the epsilon-delta definition to evaluate limits, continuity, and differentiability; Master Riemann integration and the fundamental theorems of calculus; Evaluate sequences and series of functions, focusing on pointwise and uniform convergence; Practice solving written, exam-style analysis problems designed to mimic competitive entrance tests. The course begins with core definitions of real numbers and topology, gradually advancing through limits, differentiation, integration, and function sequences. Each section pairs theoretical explanations with written practice exercises to reinforce your proof-writing skills. This program is designed for undergraduate mathematics students preparing for MSc entrance exams or anyone seeking a solid, beginner-friendly foundation in rigorous analysis. No advanced prior knowledge of real analysis is required. Start reading today to build a flawless foundation in real analysis.