📘 real analysis

1. **Problem statement:** (i) Prove that between any two real numbers $a$ and $b$ there is at least one rational number.

1. **Problem statement:** (a) Prove that between any two real numbers $a$ and $b$ there is at least one rational number.

1. **Problem Statement:** (a) Prove that between any two real numbers $a$ and $b$ there is at least one rational number.

1. **Problem:** Find the least upper bound (supremum) of the set $$A = \left\{ \frac{2}{3^n} + \frac{(-1)^n}{2^{n-1}} \mid n \in \mathbb{N} \right\}$$. 2. **Formula and rules:** Th

1. **Problem statement:** Consider the sequence $u_n = n\sqrt{n} = n^{3/2}$. We want to understand its divergence behavior and apply the definition of divergence to $u_n \in (10^6,

1. **Problem statement:** Find the supremum, infimum, maximal, and minimum elements of the set \(S = \{r \in \mathbb{Q} : 0 \leq r \leq \sqrt{2} \}\). 2. **Recall definitions:**

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

1. **Problem Statement:** Understand the Real Number System as the first unit of Real Analysis for Delhi University BSc Maths Hons syllabus. 2. **What is the Real Number System?**

1. **Problem Statement:** We have a sequence $\{a_n\}$ of real numbers such that the subsequences $\{a_{2n}\}$, $\{a_{2n-1}\}$, and $\{a_{3n}\}$ all converge. We want to show that

1. **Problem statement:** We have a sequence $\{a_n\}$ of real numbers such that the subsequences $\{a_{2n}\}$, $\{a_{2n-1}\}$, and $\{a_{3n}\}$ all converge. We need to show that

1. **Problem statement:** Given a sequence $\{a_n\}$ of real numbers, the subsequences $\{a_{2n}\}_{n=1}^\infty$ and $\{a_{2n-1}\}_{n=1}^\infty$ both converge to the same limit $L$

1. **Problem Statement:** We are given multiple questions about sets, sequences, functions, and series. We will solve the first question completely.

1. **Problem Statement:** Justify the properties in $\mathbb{R}$ using ordered field axioms: (i) Show that $x(-y) = -xy$.

1. **Problem Statement:** (a) Determine if each function $f_i : X \times X \to \mathbb{R}$ with $X=\{a,b,c,d\}$ given by the tables $f_1, f_2, f_3, f_4$ is a metric. If not, identi

1. **Problem Statement:** Find the Limit Inferior (\(\liminf\)) and Limit Superior (\(\limsup\)) of the sequences: \(a)\ (z_n) = (-2)^n\)

1. The problem asks for the value of the Dirichlet function at $x = -5$. 2. The Dirichlet function $D(x)$ is defined as:

1. **Problem Statement:** We analyze the sets $A_1, A_2, A_3, A_4, A_5$ defined as:

1. **State the problem:** We have a function $f:\mathbb{R} \to \mathbb{R}$ that is continuous at $x=\pi$ and satisfies the functional equation $$f(x+y) = f(x) + f(y)$$ for all real

1. **State the problem:** We have a function $f:\mathbb{R} \to \mathbb{R}$ that is continuous at $x=\pi$ and satisfies the functional equation $$f(x+y) = f(x) + f(y)$$ for all real

1. **Problem statement:** We analyze the properties (compactness, closedness, convexity) of given sets A, B, C and their combinations. 2. **Recall definitions:**

1. **Stating the problem:** We are given the interval $[1, 2] \subseteq \mathbb{R}$. We need to find the upper and lower bounds of this interval. Next, determine how many upper and