📘 real analysis

1. **Problem Statement:** Determine if the composition of two Riemann integrable functions is always Riemann integrable. 2. **Recall Definitions:** A function $f$ is Riemann integr

1. **State the problem:** We have the set $$A = \left\{ -1 + \frac{1}{2n + 1^2} : n \in \mathbb{N} \right\}$$ and we want to show that it is bounded. 2. **Show that $A$ is bounded:

1. **Problem statement:** We have the function $$q(x) = \begin{cases} 1 & \text{if } x \in \mathbb{Q} \\ 0 & \text{otherwise} \end{cases}$$ and we want to determine if $$q(x)$$ is

1. **Problem statement:** Construct a numerical function that is unbounded, discontinuous everywhere, but differentiable at exactly one point. 2. **Key concepts:**

1. Problem 23(a): Prove $f(x)=3x-5$ is continuous at $x=2$. 1.1. Definition: $f$ is continuous at $a$ if for every $\epsilon>0$ there exists $\delta>0$ such that $|x-a|<\delta$ imp

1. Problem 11: Show that if the sequence $x_n$ of real numbers converges to $x$ then the sequence $s_n=\frac{x_1+\cdots+x_n}{n}$ also converges to $x$. 1.1. Formula and rules: we u

1. The problem is to evaluate the expression $\sup(\{1\})$, which means finding the supremum (least upper bound) of the set containing only the number 1. 2. The supremum of a set i

1. Verify convergence of sequences using definition: (a) Show $\lim_{n \to \infty} \frac{2n+1}{5n+4} = \frac{2}{5}$.

1. **Problem statement:** We have the set $$S = \{x \in \mathbb{Q} \mid x = \frac{a}{b}, a,b \in \mathbb{Z}, 0 < b \leq 11\} \cap (-1,1)$$.

1. **Problem statement:** (a) Given functions $f, g \in R(\alpha)$ on $[a,b]$, show that $f - g \in R(\alpha)$ on $[a,b]$.

1. Statement of the problem: Prove the case of $\delta^+(f,x_0)=1$ for a real function $f$ at a point $x_0$ where $\delta^+$ denotes the right upper Dini derivative. 2. Formula and

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