Question: Let $f: \mathbb{R} \to \mathbb{R} $ be an increasing function. Suppose there are sequences $(x_n)$ and $(y_n)$ such that $x_n<0<y_n$ for all $n\geq 1$ and $f(y_n)-f(x_n) \to 0$ as $n \to \infty$. Prove that $f$ is continuous at $0$.
Attempt: We consider two arbitrary sequences $(a_n)$...
Find the the splitting field of $X^5-2$ over $\mathbb{Q}$ and find it's degree.
My approach: The roots of $X^5-2$ are $\{\sqrt[5]{2},\sqrt[5]{2}\omega,\sqrt[5]{2}\omega^2, \sqrt[5]{2}\omega^3, \sqrt[5]{2}\omega^4\}$ where $\omega=e^{2\pi i/5}$.
It's quite easy to show that splitting field of $X...
A palindromic number or numeral palindrome is a number that remains the same when its digits are reversed. Like 16461, for example, it is "symmetrical". The term palindromic is derived from palindrome, which refers to a word (such as rotor or racecar) whose spelling is unchanged when its letters are reversed. The first 30 palindromic numbers (in decimal) are:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 111, 121, 131, 141, 151, 161, 171, 181, 191, 202, … (sequence A002113 in the OEIS).Palindromic numbers receive most attention in the realm of recreational mathematics...
You're right $2|5$ is a false statement. Here are some true statements using |, see if you can see why they are true:
$$a|b\wedge b|c\Longrightarrow a|c\\\{n|ab\Longrightarrow n|a \vee n|b\}\Longleftrightarrow \text{$n$ is prime}$$
Here is an example from the book Groups, Graphs, and Trees (Ignore example 1.15; I accidently included it in my snippet of the pdf):
From my understanding, the symmetry group of a graph $\Gamma$ consists of all bijections $f : V(\Gamma) \to V(\Gamma)$ such that $v,w$ are adjacent vertices if a...
