Intereting Posts

Cohomology with Coefficients in the sheaf of distributions
Proof that a sum of the first period of powers of integer roots of rationals is irrational
Upper bound on the number of charts needed to cover a topological manifold
The p-adic numbers as an ordered group
Show that $\int_0^\pi \log^2\left(\tan\frac{ x}{4}\right)dx=\frac{\pi^3}{4}.$
Using the Uniform Continuity of the Characteristic Function to Show it's Differentiable
Showing that $\lnot Q \lor (\lnot Q \land R) = \lnot Q$ without a truth table
difference between parallel and orthogonal projection
Repeatedly taking mean values of non-empty subsets of a set: $2,\,3,\,5,\,15,\,875,\,…$
Population Dynamics model
Finding binomial coefficients of product of two binomials
Definition of functional derivative
Degree of field extension
If the index $n$ of a normal subgroup $K$ is finite, then $g^n\in K$ for each $g$ in the group.
holomorphic function on punctured disk satisfying $\left|f\left(\frac{1}{n}\right)\right|\leq\frac{1}{n!}$ has an essential singularity at $0$

Can anyone help me solve this question? I’ve been working on it for two days already.

Prove that for real numbers $x$ and $y$ with $x < y$, there is a rational and an

irrational between $x$ and $y$ in the following case when $x < y \le 0$.

Since $y \le 0$ is logically equivalent to $-y\ge 0$, we can use the squeezing in theorem to show that a rational $r$, and an irrational $q$ exist in the interval $(0,-y)$ which is the same as saying that there is a rational and an irrational in the interval $(y,0)$.

- Showing that $1^k+2^k + \dots + n^k$ is divisible by $n(n+1)\over 2$
- Verifying proof :an Ideal $P$ is prime Ideal if $R/P$ is an integral domain.
- Prove that $\int_0^\pi\frac{\cos x \cos 4x}{(2-\cos x)^2}dx=\frac{\pi}{9} (2160 - 1247\sqrt{3})$
- True or False $A - C = B - C $ if and only if $A \cup C = B \cup C$
- Proving statement - $(A \setminus B) \cup (A \setminus C) = B\Leftrightarrow A=B , C\cap B=\varnothing$
- Interspersing of integers by reals

For the second part, I want to show that there exists a rational and irrational in the interval $(x,y)$. Since $x < y$ is logically equivalent to $-y<-x$, if we show that there is a rational and an irrational between $(-y,-x)$ then we’re done. To show this I’m trying to use Archimedean Principle that there exists a rational $mr> -y$ and I am stuck here.

- Recursive Monotone Decreasing Sequence Proof ${x_{k}} = \frac{1}{2}\left({x_{k-1}+\frac{a}{{x_{k-1}}}}\right)$
- Limit points of particular sets of real numbers.
- What does it mean for rational numbers to be “dense in the reals?”
- How to prove these inequalities: $\liminf(a_n + b_n) \leq \liminf(a_n) + \limsup(b_n) \leq \limsup(a_n + b_n)$
- Singular continuous functions
- If a function is undefined at a point, is it also discontinuous at that point?
- Does There exist a continuous bijection $\mathbb{Q}\to \mathbb{Q}\times \mathbb{Q}$?
- Continuity of $\arg (z)$
- Clarifying question on meaning of derivative of a linear transformation
- Archimedean property

Assume $y-x>0$. By the Archimedean property, there is $n$ such that $n(y-x)=ny-nx>1$. But if two reals have a gap $>1$; there must be an integer between them. So…?

For irrationals, pick your favourite one in $(0,1)$ and transport it between two rationals by an appropriate map. That is, if $\kappa$ is irrational, and $r<s$ rationals, can you find rationals $\alpha\neq 0 ,\beta$ such that $r<\alpha\kappa+\beta<s$? Then use the first part to conclude. Note $\alpha\kappa+\beta$ is still irrational! (Why?)

- What happens if we remove the requirement that $\langle R, + \rangle$ is abelian from the definition of a ring?
- If $X$ is symmetric, show $k(X^2)$ = $k(X)^2$
- Moebius band not homeomorphic to Cylinder.
- Prove that projection operator is non-expansive
- Define integral for $\gamma,\zeta(i) i\in\mathbb{N}$ and Stirling numbers of the first kind
- If $G$ is non-abelian group of order 6, it is isomorphic to $S_3$
- Set up double integral of ellipse in polar coordinates?
- How many numbers are in the Fibonacci sequence
- Infinite group with only two conjugacy classes
- About a measurable function in $\mathbb{R}$
- Why isn't there a contravariant derivative? (Or why are all derivatives covariant?)
- How do we take second order of total differential?
- Archimedes' derivation of the spherical cap area formula
- curve of constant curvature on unit sphere is planar curve?
- Choosing a contour to integrate over.