Intereting Posts

Is there a real-valued function $f$ such that $f(f(x)) = -x$?
Calculate the viewing-angle on a square (3d-calc)
Is there such thing as an imaginary (imaginary number)?
A Counter Example about Closed Graph Theorem
integrate $\int \frac{1}{e^{x}+e^{ax}+e^{a^{2}x}} \, dx$
$1^n +2^n + \cdots +(p-1)^n \mod p =$?
Prove the inequality $x+\frac{1}{x}\geq 2$?
book for metric spaces
Cardinality of the Union is less than the cardinality of the Cartesian product
Minimum and Maximum eigenvalue inequality from a positive definite matrix.
Invertibility of compact operators in infinite-dimensional Banach spaces
Seifert-van-Kampen and free product with amalgamation
Test symmetricity for a sparse matrix
The value of $\sum_{k=0}^\infty k^n x^k$
Finding an asymptotic for the sum $\sum_{p\leq x}p^m$

The distance between a point $a \in \mathbb{R}$ and a set $X \subset \mathbb{R}$ is defined as $$d(a,X) := \inf\{|x-a|: x \in X\}.$$ How to prove if $X$ is closed, then there is a $b \in X$ such that $d(a,X) = |b-a|$?

I’ve constructed a decreasing sequence converging to $d$ as follows: Given $r > d(a,X)$, there is a $x \in X$ such that $|x-a| < r$. Repeating the process with $r_{n+1} := \frac{d+r_n}{2}$ we get the inequality:

$$d \leq |x_n-a| < r_n$$

- Infinite Sum Calculation: $\sum_{k=0}^{\infty} \frac{1}{(2k+1)^2} = \frac{3}{4} \sum_{n=1}^{\infty} \frac{1}{n^2}$
- Is tensor product of Sobolev spaces dense?
- Show that $f(x)=g(x)$ for all $x \in \mathbb R$
- Bartoszyński's results on measure and category and their importance
- Continuous function preserving rational difference.
- preservation of extreme points under linear transformation

It’s easy to prove that $r_n \mapsto d$, and therefore $|x_n-a| \mapsto d$. If i could show the set $A := \{|x-a|: x\in X\}$ is closed, the result would be immediate. This is somehow my second question, is true that for every closed set $X$, the set $|X| := \{|x|: x\in X\}$ is closed?

Be free to contribute alternative proofs, i would appreciate.

- Why do we need min to choose $\delta$?
- Is the determinant differentiable?
- Infinite metric space has open set $U$ which is infinite and its complement is infinite
- How to prove that $\mu=0$
- A continuous function that attains neither its minimum nor its maximum at any open interval is monotone
- Equivalent metrics
- On the existence of a certain sequence of positive numbers
- Can you prove this property?
- How to show that the set of all Lipschitz functions on a compact set X is dense in C(X)?
- Finding an example of nonhomeomorphic closed connected sets

**Hint:** Pick a $b \in X$. Then, it is enough to look only to the set $Y:= \{ x \in X | d(a, x) \leq d(a, b) \}$.

Then $Y= X \cap B_{ d(a, b)}(a)$, where the second set is the closed ball.. Now, $Y$ is closed and bounded thus compact…

Can you prove that there exists a $y \in Y$ so that $d(a,y)= d(a, Y)$? Keep in mind that now you have compactness instead of closure….

**To complete the proof**

Let $d =d(a, X)=d(a,Y)$. Then for each $n$ you can find some $x_n \in Y$ so that $d \leq d(a,x_n) \leq d+\frac{1}{n}$.

The sequence $x_n \subset Y$ must have a cluster point $y \in Y$, since $Y$ is compact.

**Question:** What is $d(a,y)$?

Another way to look at this: Letting $r$ be sufficiently large, $d(a,X) = d(a, X \cap B(0,r))$, where $B(0,r)$ is the closed ball of radius $r$ centered at the origin. Use the triangle inequality to show that $|a – x|$ is a continuous function of $x$ for $x \in X \cap B(0,r)$. Since $ X \cap B(0,r)$ is compact, it achieves its minimum value at some $b \in X \cap B(0,r)$, and this $b$ will minimize $|a – b|$ over all $b \in X$ as well since $d(a,X) = d(a, X \cap B(0,r))$.

Yet another way: Let $E_n = \{x \in X: |x – a| \leq d(a,X) + {1 \over n}\}$. The $E_n$’s are nested compact sets and thus have nonempty intersection i.e. you can choose $b \in \cap_n E_n$. Then since $d(a,X) \leq |a – b| \leq d(a,X) + {1 \over n}$ for all $n$, you must have $|a – b| = d(a,X)$.

- How is a set subset of its power set?
- GRE problem involving LCD, prime factorization, and sets.
- The Instant Tangent
- Prove that $f$ continuous and $\int_a^\infty |f(x)|\;dx$ finite imply $\lim\limits_{ x \to \infty } f(x)=0$
- Example of different topologies with same convergent sequences
- What Is The Limit Of The Sequence: $\frac{n^3}{{((3n)!)^\frac{1}{n}}}$
- How to effectively study math?
- A semicontinuous function discontinuous at an uncountable number of points?
- Factoring with fractional exponents
- Example of a boundary point that is not simple
- $\inf_{x\in}f(x)=\inf_{x\in\cap\mathbb{Q}}f(x)$ for a continuous function $f:\to\mathbb{R}$
- Fundamental group of quotient of $S^1 \times $
- clock related challenge
- Prove that if $n^2$ is divided by 3, then also $n$ can also be divided by 3.
- Solving an equation with irrational exponents