Intereting Posts

The PDF of the area of a random triangle in a square
Expectation value of a product of an Ito integral and a function of a Brownian motion
On radial limits of Blaschke Products
If $K^{\mathrm{Gal}}/F$ is obtained by adjoining $n$th roots, must $K/F$ be as well?
Radon-Nikodym derivative of product measure
showing that $\lim_{x\to b^-} f(x)$ exists.
When does the $f^{(n)}$ converge to a limit function as $n\to\infty$?
Show that number of solutions satisfying $x^5=e$ is a multiple of 4?
Prove $\frac{1}{a^3+b^3+abc}+\frac{1}{a^3+c^3+abc}+\frac{1}{b^3+c^3+abc} \leq \frac{1}{abc}$
Norm of the product of two regular ideals of an order of an algebraic number field
Is this equality true or it is not necessarily true?
Do dynamic programming and greedy algorithms solve the same type of problems?
Intuitive use of logarithms
Proving an operator is compact exercise
How many elements are there in the group of invertible $2\times 2$ matrices over the field of seven elements?

Why is this true?

I think I can find a countable union of compact sets $\cup_{k=1}^\infty X_k$ such that $\cup X_k \subseteq U$ and the lebesgue measure of $U \setminus \cup X_k$ is zero.

(for any $k\in \mathbb{N}$, we can find a closed set $Y_k \subset U$ such that $\lambda(U\setminus Y_k)<\frac{1}{k}$. (Take $X_k=B(k)\cap Y_k$ where $B(k)$ is the ball of radius $k$.)

- $(X,\mathscr T)$ is compact $\iff$ every infinite subset of $X$ has a complete limit point in $X$.
- Proving a necessary and sufficient condition for compactness of a subset of $\ell^p$
- Metrizable topological space $X$ with every admissible metric complete then $X$ is compact
- Compact space and Hausdorff space
- Arzela-Ascoli and compactness in $C(X), l^p, L^p$
- Is a discrete set inside a compact space necessarily finite?

But that doesn’t solve the problem.

- Is the outer boundary of a connected compact subset of $\mathbb{R}^2$ an image of $S^{1}$?
- Uniform convergence of functions, Spring 2002
- Continuous mapping on a compact metric space is uniformly continuous
- Proving that the closure of a subset is the intersection of the closed subsets containing it
- Showing one point compactification is unique up to homeomorphism
- Examples of fundamental groups
- Viewing Homotopies as Paths in $\mathcal{C}^0(X,Y)$
- A valid proof for the invariance of domain theorem?
- What are the components and path components of $\mathbb{R}^{\omega}$ in the product, uniform, and box topologies?
- Are $L^\infty$ bounded functions closed in $L^2$?

Let

$$X_k:=\left\{x\in \mathbb R^n,\left\lVert x\right\rVert\leqslant k\}\cap \{x,d\left(x,U^c\right)\geqslant k^{-1}\right\}.$$

Then $X_k$ is closed (as an intersection of such sets) and bounded in $\mathbb R^n$ hence compact. We have $U=\bigcup_{k\geqslant 1}X_k$. Indeed, by definition $X_k\subset U$ for all $k$, and if $x\in U$, we can find $n$ such that

$\lVert x\rVert\leqslant n$. As $U^c$ is closed and $x\notin U^c$, $d\left(x,U^c\right)$ is positive. So take $k$ such that $d\left(x,U^c\right)\geqslant k^{-1}$ and $N:=\max\left\{n,k\right\}$. Then $x$ belongs to $X_N$.

- An example of two infinite-dimensional vector spaces such that $\dim_{\mathbb{F}}\mathcal{L}(U,V)> \dim_{\mathbb{F}}U\cdot \dim_{\mathbb{F}}V$
- Find all $a,b,c\in\mathbb{Z}_{\neq0}$ with $\frac ab+\frac bc=\frac ca$
- Compute $\sum \frac{1}{k^2}$ using Euler-Maclaurin formula
- Eisenstein Criterion with a twist
- integral of $\int \limits_{0}^{\infty}\frac {\sin (x^n)} {x^n}dx$
- Proving an operator is compact exercise
- Block inverse of symmetric matrices
- for two positive numbers $a_1 < b_1$ define recursively the sequence $a_{n+1} = \sqrt{a_nb_n}$
- Understanding the differential $dx$ when doing $u$-substitution
- Why is pointwise continuity not useful in a general topological space?
- Suppose $(s_n)$ converges and that $s_n \geq a$ for all but finitely many terms, show $\lim s_n \geq a$
- Exercise 1 pg 33 from “Algebra – T. W. Hungerford” – example for a semigroup-hom but not monoid-hom..
- What does “IR” mean in linear algebra?
- Forming Partial Fractions
- Properties of Hardy operator $T(u)(x)=\frac{1}{x}\int_0^x u(t)dt$