📘 real analysis

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

1. The problem asks to find the supremum (least upper bound) and infimum (greatest lower bound) of the set $$S=\left\{\frac{n-m}{n+m}: n,m \in \mathbb{N}\right\}$$ where $n$ and $m

1. The problem is to show that the set $ (1,2) $ has no maximum element. 2. The set $ (1,2) $ is an open interval, meaning it includes all real numbers between 1 and 2 but does not

1. First, state the inequalities to prove: a. For all real numbers $x, y$, prove that $|x| + |y| \leq |x + y| + |x - y|$.

1. **Show the inequalities in Exercise 2:** **a.** Show that $|x|+|y| \leq |x+y|+|x-y|$ for all $x,y \in \mathbb{R}$.