Intereting Posts

Proving that the smooth, compactly supported functions are dense in $L^2$.
Exercise from Stein with partial differential operator
Plane intersecting line segment
Cardinality of the set of differentiable functions
Is a coordinate system a requirement for a vector space?
Is $x_{n+1}=\frac{x_n}{2}-\frac{2}{x_n}$ bounded?
In categorical terms, why is there no canonical isomorphism from a finite dimensional vector space to its dual?
Verifying Carmichael numbers
Show that $2 xy < x^2 + y^2$ for $x$ is not equal to $y$
Is a line parallel with itself?
Using Fermat's little theorem to find remainders.
Is $72!/36! -1$ divisible by 73?
Group presentation for semidirect products
Is there an irrational number containing only $0$'s and $1$'s with continued fraction entries less than $10$?
How to prove that $\frac{x^2}{yz+2}+\frac{y^2}{zx+2}+\frac{z^2}{xy+2}\geq \frac{x+y+z}{3}$ holds for any $(x,y,z)\in^3$

This is from a review packet:

Let $d:\mathbb{R} \to \mathbb{R}$ be defined as $$d(x,y)=\frac{|x-y|}{1+|x-y|}.$$

i) Show that $(\mathbb{R},d)$ is a bounded metric space.

ii) Show that $A=[a,\infty)$ is a closed and bounded subset of $\mathbb{R}$.

iii) Show that $A=[1,\infty)$ is not compact.

- Proving $\frac{1}{\sin^{2}\frac{\pi}{14}} + \frac{1}{\sin^{2}\frac{3\pi}{14}} + \frac{1}{\sin^{2}\frac{5\pi}{14}} = 24$
- for infinite compact set $X$ the closed unit ball of $C(X)$ will not be compact
- Is $dx\,dy$ really a multiplication of $dx$ and $dy$?
- Evaluating sums using residues $(-1)^n/n^2$
- $\lim_{x\to 0+}\ln(x)\cdot x = 0$ by boundedness of $\ln(x)\cdot x$
- A function satisfy $x\frac{\partial f(x,y)}{\partial x}+y\frac{\partial f(x,y)}{\partial y}=0$ in a convex domain implies it is a constant

For (i) I think this is a true observation: $d(x,y)\leq1$ for an arbitrary $x,y \in \mathbb{R}$. I’m not sure where to go from there however.

For (ii) and (iii) – I’m assuming those will following quickly from (i).

- Closed subset of compact set is compact
- Compactly supported continuous function is uniformly continuous
- Fourier transform of the derivative - insufficient hypotheses?
- Uncountable set with exactly one limit point
- Does $\sum_{j = 1}^{\infty} \sqrt{\frac{j!}{j^j}}$ converge?
- Infinite Product $\prod\limits_{k=1}^\infty\left({1-\frac{x^2}{k^2\pi^2}}\right)$
- Prove that $C^1()$ with the $C^1$- norm is a Banach Space
- Uniformly distributed rationals
- If $f,g$ are both analytic and $f(z) = g(z)$ for uncountably many $z$, is it true that $f = g$?
- Accumulation points of $\{ \sqrt{n} - \sqrt{m}: m,n \in \mathbb{N} \}$

Since $d(x,y)\leq 1$ for $\forall x \neq y$ we get that $\mathbb R,[a,+\infty)\subset B_1(0)$. Finally $[1,+\infty)$ is not compact since the open cover $\{(0,n), \ n \in \mathbb{N}\}$ does not have a finite subcover of $[1,+\infty)$.

- How likely is it for a randomly picked number to be larger than all previously chosen numbers?
- Is there a simple, constructive, 1-1 mapping between the reals and the irrationals?
- How many faces of a solid can one “see”?
- DTFT of a triangle function in closed form
- Find $\lim\limits_{x\to 0}\frac{\sqrt{2(2-x)}(1-\sqrt{1-x^2})}{\sqrt{1-x}(2-\sqrt{4-x^2})}$
- How do I sum two Poisson processes?
- Projections: Ordering
- Calculate Third Point of Triangle
- WLOG means losing generality?
- Cube stack problem
- Infinitesimals – what's the intuition?
- All the ternary n-words with an even sum of digits and a zero.
- making mathematical conjectures
- A mouse leaping along the square tile
- From $\sigma$-algebra on product space to $\sigma$-algebra on component space