Real Analysis | Last 5 Years PYQ Discussion | CSIR NET | MH SET | RJ SET | Lec-05 | IFAS

Real Analysis | Last 5 Years PYQ Discussion | CSIR NET | MH SET | RJ SET | Lec-05 | IFAS explained with clear concepts and exam-focused approach. Master crucial Real Analysis PYQs with problem‑solving tricks and concept clarity. This session covers CSIR NET, MH SET, and RJ SET exam patterns, emphasizing uncountable sets, continuity, differentiability, and monotonic functions — all backed by step‑wise solutions from last 5 years. 🔥Join India's Best Coaching for CSIR NET and GATE Exams: https://ifasonline.com/csir-net/mathe... 📞 Need Guidance? Call/WhatsApp: 9172266888 ━━━━━━━━━━━━━━━━━━━━━━━━━ ⏳ Lecture Timeline: [00:00:00] – Introduction to Real Analysis [00:03:57] – Question 1: Uncountability of Open & Connected Subsets [00:11:18] – Question 2: Existence of Bounded / Monotonic Subsequence [00:11:50] – Important: Every sequence has a monotonic subsequence [00:12:13] – Question 3: Continuous Map between [0,1] and (0,1) [00:16:13] – Concept: Constructing continuous map (0,1) → [0,1] onto [00:18:35] – Question 4: Properties of f(x) = x sin(1/x) [00:20:15] – Concept: Continuity & Riemann Integrability on [0,1] [00:24:35] – Question 5: Supremum of Union of Bounded Sets [00:28:49] – Important: sup(Z) = max{sup X, sup Y} [00:29:53] – Question 6: Intersection of Families of Sets (Empty/Non‑empty) [00:30:45] – Result: Only the intersection of Jₙ is non‑empty [00:39:37] – Question 7: Inequality |sin x – sin y| ≤ |x – y| [00:40:21] – Derivation: Trig identity & |sin θ| ≤ |θ| [00:42:35] – Question 8: Limit Superior of a Sequence [00:47:22] – Result: lim sup = 1 [00:47:52] – Question 9: Integral Equation ∫₀ˣ f = ∫ₓ¹ f [00:48:44] – Method: Differentiate both sides [00:50:31] – Result: f ≡ 0 [00:54:27] – Complete Session (Real Analysis) ━━━━━━━━━━━━━━━━━━━━━━━━━ 📖 Topics Covered in This Lecture: • Uncountable open and connected subsets PYQ tricks • Sequences: bounded subsequence, limit superior concepts • Continuous maps from compact to open intervals • x sin(1/x) differentiability and integrability problems • Supremum, maximum, and intersection of nested intervals • sin x – sin y inequality using mean‑value relation • Integral equation leading to zero function • Strictly vs non‑strictly monotonic functions PYQ ━━━━━━━━━━━━━━━━━━━━━━━━━ 📚 Lecture Summary: In this session, IFAS Expert Faculty focuses on Real Analysis PYQ discussions from CSIR NET, MH SET, and RJ SET, breaking down complex statements about uncountable sets, connected subsets, and continuous mappings. The instructor solves problems on sequences, limit superior, function x sin(1/x), and supremum‑maximum distinctions with rigorous reasoning. Key tricks include identifying bounded subsequences and using integral conditions to determine constant zero functions. The session reinforces Real Analysis exam strategies through direct, repeatable problem‑solving techniques for future papers. ━━━━━━━━━━━━━━━━━━━━━━━━━ 🎓 This lecture is highly beneficial for students preparing for: ✅ CSIR NET Mathematical Science ✅ GATE Mathematics ✅ SET Mathematics ✅ TIFR Mathematics ✅ PhD Entrance Exams ━━━━━━━━━━━━━━━━━━━━━━━━━ 🏆 Why IFAS is India’s No. 1 Choice for CSIR NET & GATE Mathematics? (Our Results Speak): ✅ CSIR NET Dec 2025: 4000+ Selections (AIR 3, 8, 9.....) ✅ CSIR NET June 2025: 4200+ Selections (AIR 1, 2, 6, 10...) ✅ CSIR NET Dec 2024: 4000+ Selections (AIR 5, 6, 7, 8, 9...) ✅ CSIR NET June 2024: 2000+ Selections (AIR 3, 10, 13...) ✅ CSIR NET Dec 2023: 1200+ Selections (AIR 2, 4, 5...) 📄 View IFAS Result Sheet: 👉 https://drive.google.com/file/d/1THh6... ━━━━━━━━━━━━━━━━━━━━━━━━━ 🔗 Important Links: 🌍 Visit our Website: https://ifasonline.com 📞 Contact for Queries: 9172266888 📘 Free Counselling: https://ifasonline.com/csir-net/mathe... ━━━━━━━━━━━━━━━━━━━━━━━━━ 🔗 India's Best Online Classes and Books: 🔥CSIR NET Mathematical Science Live & Recorded Course: https://ifasonline.com/csir-net/mathe... 🎯 GATE Mathematics Recorded Course: https://ifasonline.com/gate/mathemati... 📘CSIR NET Mathematical Science Books: https://ifasonline.com/csir-net/mathe... ━━━━━━━━━━━━━━━━━━━━━━━━━ 📲 Download IFAS App: Android: https://play.google.com/store/apps/de... IOS: https://apps.apple.com/us/app/ifas/id... ━━━━━━━━━━━━━━━━━━━━━━━━━ Stay Updated Via Other Social Media: ✔Telegram: https://t.me/NET_GATE_SET_Maths_Discu... ✔Instagram:   / csir_net_mathematical_science   #realanalysis #mahasangram #pyqdiscussion #csirnetmathematics #examstrategy #csirnetmathematicalscience #gatemathematics #csirnetmaths #csirnetjune2026 #csirnetlecture #ifasmathematics