Intereting Posts

Prove that $ AA^T=0\implies A = 0$
Solve $\lim_{x\to 0} \frac{\sin x-x}{x^3}$
Difference between the Jacobian matrix and the metric tensor
Does one necessarily need an MS in Math before taking a PhD in Math?
Two different formulas for standard error of difference between two means
A “non-trivial” example of a Cauchy sequence that does not converge?
Prove that convex function on $$ is absolutely continuous
Show that $\int_0^1\ln(-\ln{x})\cdot{\mathrm dx\over 1+x^2}=-\sum\limits_{n=0}^\infty{1\over 2n+1}\cdot{2\pi\over e^{\pi(2n+1)}+1}$ and evaluate it
Maximum value of multiplicative order
Subspaces of separable normed spaces
Prove that $\frac{a_1^2}{a_1+a_2}+\frac{a_2^2}{a_2+a_3}+ \cdots \frac{a_n^2}{a_n+a_1} \geq \frac12$
Describe the topology of Spec$(\mathbb{R})$
If a player is 50% as good as I am at a game, how many games will it be before she finally wins one game?
Let $a,b$ be positive integers such that $a\mid b^2 , b^2\mid a^3 , a^3\mid b^4 \ldots$ so on , then $a=b$?
“Canceling Out The Zeroes” In A Mathematically Sane Way $\frac{0\times x}{0\times 1}$

The theorem:

Let $C$ be a non-empty closed convex subset of a Hilbert space

$X$, and let $x \in X$. Then there exists a unique $y_0 \in C$ such that $||x-y_0|| \le ||x – y||$ all $y \in C$. In other words there is a point in $C$ which is closest to $x$.

I’m reading a proof here (https://www0.maths.ox.ac.uk/system/files/coursematerial/2014/3075/48/15B4.2-webnotes-all.pdf, p.4) and have two questions about the initial steps in the prooof (I can follow the later steps.):

- ZF and the Existence of Finitely additive measure on $\mathcal{P}(\mathbb{R})$
- Axiom of Choice and Determinacy
- Zorn's Lemma and Injective Modules
- picking a witness requires the Axiom of Choice?
- Mathematical questions whose answer depends on the Axiom of Choice
- Vector space bases without axiom of choice

- “Let $d = inf\{||x – y||: y \in C\}$” I think that the infimum should exist by the property (axiom) of completeness of the real numbers. $\{||x – y||: y \in C\}$ is a set of real numbers which is bounded below (all norms are $\ge 0$) and therefore has a greatest lower bound – is this correct ?
- “Let $(y_n)$ be a sequence of points in C such that $||x – y_n|| \to d$”. I think such a sequence should exist by the following reasoning: With $d = inf\{||x – y||: y \in C\}$ then for any $\epsilon > 0$ there must be a point $y$ in $C$ such that $||x – y|| \lt d + \epsilon$ (otherwise $d$ is not the infimum). If for this point $||x – y|| = d $ then $y_0 = y$ and we are finished: $(y_0)$ is a one element convergent sequence. Otherwise $||x – y|| = d + \epsilon`$ where $\epsilon` \le \epsilon$ and I can repeat the process to find a
**different**point where $||x – y|| \lt d + \epsilon`/2$ and so generate a convergent sequence. I’m not sure this is valid, and it occurs to me that unless $C = \{y_0\}$ there are probably an infinite number of points at each step and the Axiom of choice is then required to extract a convergent sequence. Any help would be appreciated.

- Does the law of the excluded middle imply the existence of “intangibles”?
- Absolute Continuity of Finite Borel Measure Characterized by Orthonormal Basis
- Spectral Measures: Support vs. Spectrum
- Axiom of Choice and Right Inverse
- how to show $f$ attains a minimum?
- Orthonormal basis in $L^2(\Omega)$ for bounded $\Omega$
- Non-aleph infinite cardinals
- What is the dual space in the strong operator topology?
- Unprovable statements in ZF
- Finite choice without AC

1) You are correct. Every set that is bounded below in $\mathbb{R}$ has a least lower bound.

2)The standard why to show that there exists such a sequence $(y_n)$ is as follows. Let $n\in\mathbb{N}$ then as d is the infimium, $d+\frac{1}{n}$ is not a lower bound so there exists $y_n$ such that $d<||x-y_n||<d+\frac{1}{n}$. This gives a sequence of points $(y_n)$. Then it should be obvious that $||x-y_n||\rightarrow d$.

- Prove that if $n$ is a composite, then $2^n-1$ is composite.
- If the integral of $c/x$ is $c.log(x)+C$ what is the base?
- What's an intuitive way to think about the determinant?
- $x,y$ are real, $x<y+\varepsilon$ with $\varepsilon>0$. How to prove $x \le y$?
- Prove: $(a + b)^{n} \geq a^{n} + b^{n}$
- Prove $ 1 + 2 + 4 + 8 + \dots = -1$
- How find this sum $I_n=\sum_{k=0}^{n}\frac{H_{k+1}H_{n-k+1}}{k+2}$
- How to prove $\forall Y \subset X \ \forall U \subset X \ (Y \setminus U = Y \cap (X \setminus U))$?
- Why does $\lim_{n\to\infty} z_n=A$ imply $\lim_{n\to\infty}\frac{1}{n}(z_1+\cdots+z_n)=A$?
- What is the probability that $X<Y$?
- transcendental entire function, $Aut(\mathbb{C})$
- Asymmetric Random Walk / Prove $E = \frac{b}{p-q}$ / How do I use hint?
- Square Root Inequality
- Prove that $K$ is isomorphic to a subfield of the ring of $n\times n$ matrices over $F$.
- Non-Noetherian ring with a single prime ideal