1. **Problem Statement:** Understand the concepts of the real number system, supremum, and infimum as per the Delhi University BSc Maths Hons syllabus for Real Analysis 1st unit. 2. **Real Number System:** The real numbers include all rational and irrational numbers. They form a complete ordered field, meaning every non-empty set of real numbers that is bounded above has a least upper bound (supremum), and every non-empty set bounded below has a greatest lower bound (infimum). 3. **Supremum (Least Upper Bound):** For a set $S \subseteq \mathbb{R}$, an upper bound is a number $u$ such that $x \leq u$ for all $x \in S$. The supremum, denoted $\sup S$, is the smallest such upper bound. It may or may not be in the set. 4. **Infimum (Greatest Lower Bound):** Similarly, a lower bound $l$ satisfies $l \leq x$ for all $x \in S$. The infimum, denoted $\inf S$, is the greatest such lower bound. It may or may not be in the set. 5. **Important Properties:** - If the supremum is in the set, it is the maximum. - If the infimum is in the set, it is the minimum. - Every non-empty set bounded above has a supremum. - Every non-empty set bounded below has an infimum. 6. **Example:** Consider the set $S = \{x \in \mathbb{R} : 0 < x < 1\}$. - Upper bounds include 1, 2, 100, etc. - The supremum is $\sup S = 1$ (smallest upper bound). - Lower bounds include 0, -1, -10, etc. - The infimum is $\inf S = 0$ (greatest lower bound). - Note: Neither 0 nor 1 is in $S$, so $S$ has no minimum or maximum. 7. **Summary:** Supremum and infimum help us understand bounds of sets in the real number system, crucial for limits, continuity, and other analysis concepts. This explanation aligns with the Delhi University BSc Maths Hons syllabus and is designed for easy understanding before exams.