Complete Real Analysis Syllabus Breakdown
Real Analysis is typically divided into core modules that build upon each other. Understanding this structure is crucial for systematic preparation.
Module 1: Real Number System & Sequences
- Properties of Real Numbers (Completeness, Archimedean Property)
- Sequences: Convergence, Divergence, Boundedness
- Monotone Sequences & Monotone Convergence Theorem
- Subsequences & Bolzano-Weierstrass Theorem
- Cauchy Sequences & Cauchy Convergence Criterion
- Limit Superior and Limit Inferior
- Series: Convergence Tests (Comparison, Ratio, Root, Integral)
Module 2: Limits & Continuity
- Limit of a Function (ε-δ definition)
- Sequential Criterion for Limits
- Continuity at a Point and on an Interval
- Types of Discontinuities
- Algebra of Continuous Functions
- Uniform Continuity & its Properties
- Intermediate Value Theorem & Applications
- Extreme Value Theorem
Module 3: Differentiation
- Derivative: Definition & Basic Properties
- Differentiability implies Continuity (converse false)
- Mean Value Theorems (Rolle's, Lagrange's, Cauchy's)
- Taylor's Theorem with Remainder
- L'Hospital's Rule
- Higher Order Derivatives
- Convex Functions & Derivatives
- Darboux's Theorem (Intermediate Value Property for Derivatives)
Module 4: Riemann Integration
- Partitions, Upper & Lower Sums
- Riemann Integrability Criteria
- Properties of Riemann Integral
- Integrability of Continuous & Monotone Functions
- Fundamental Theorems of Calculus
- Integration Techniques & Substitution
- Improper Integrals: Convergence Tests
- Functions of Bounded Variation
Module 5: Sequences & Series of Functions
- Pointwise & Uniform Convergence
- Weierstrass M-Test
- Properties Preserved Under Uniform Convergence
- Power Series & Radius of Convergence
- Taylor Series & Analytic Functions
- Fourier Series (Introduction)
- Metric Spaces: Basic Concepts
Module 6: Metric Spaces & Topology
- Definition & Examples of Metric Spaces
- Open & Closed Sets in Metric Spaces
- Compactness: Definitions & Properties
- Heine-Borel Theorem (for ℝⁿ)
- Connectedness & Path Connectedness
- Complete Metric Spaces
- Contraction Mapping Theorem
Important Note:
The weightage distribution varies across universities. Our teaching adapts to your specific syllabus pattern. We provide customized study plans based on past 10 years' question papers of your university.
How Sourav Sir's Classes Helps Students Excel
Our teaching methodology is specifically designed to overcome the challenges students face in Real Analysis.
Conceptual Clarity First
We begin with intuitive explanations using visual aids and real-world analogies before introducing formal definitions. This builds strong foundational understanding.
Interconnected Learning
We show how different concepts in Real Analysis connect to each other and to other mathematical domains, creating a cohesive mental framework.
Our Unique Teaching Approach
Live Interactive Classes
Real-time doubt resolution with personalized attention. Small batch sizes ensure every student gets individual guidance.
Structured Study Material
Comprehensive notes with color-coded important theorems, proofs, and examples. Summarized cheat sheets for quick revision.
Graded Problem Sets
From basic to advanced level problems. Regular assignments with detailed feedback on proof-writing techniques.
Recorded Lectures
Access to all class recordings for revision. Slow-motion explanation of complex proofs and theorem applications.
Progress Tracking
Regular assessments with detailed performance analytics. Identification of weak areas with customized improvement plans.
Exam Strategy Sessions
Time management techniques for exams. Question selection strategy and mark maximization approaches.
Proof-Writing Mastery Program
Our unique 6-step approach to proof writing: 1) Understanding the statement, 2) Identifying known facts, 3) Choosing proof technique, 4) Structuring the argument, 5) Writing clearly, 6) Checking for gaps. Weekly proof-writing workshops with individual feedback.
Student Success Stories
Here's what our students have achieved with our Real Analysis program:
How to Score 90+ Marks in Real Analysis
Based on our analysis of top scorers across universities, here's the winning formula:
Exam Strategy Breakdown
Time Allocation Strategy
Allocate time proportionally to marks: 50% of time for 70% of easier questions, 50% for remaining 30% challenging proofs.
Answer Presentation
Neat formatting, clear step-by-step proofs, highlighting key theorems used. Well-presented answers can earn up to 15% extra marks.
Selective Studying
Focus on high-weightage topics identified from past 10 years' papers. We provide topic-wise importance analysis for each university.
Proof Verification
Always verify if converse of a theorem is true (often asked). Prepare counterexamples for common misconceptions.
30-Day Revision Plan for Finals
- Days 1-10: Revise all definitions, theorems, and statements (without proofs)
- Days 11-20: Master proofs of important theorems (we identify which ones)
- Days 21-25: Solve previous 10 years' question papers (timed conditions)
- Days 26-28: Weak area strengthening based on test performance
- Days 29-30: Final revision of theorems, definitions, and important counterexamples
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