OpenLLM-France/wikimedia huggingface dataset:
# PlanetPhysics/Harmonic Series

The harmonic series $\sum _{k=1}^{\infty }{\frac {1}{k}}\;=\;1+{\frac {1}{2}}+{\frac {1}{3}}+\ldots$ satisfies the necessary condition of convergence $\lim _{k\to \infty }a_{n}\;=\;0$ for the series \,$a_{1}+a_{2}+a_{3}+\ldots$ of real or complex terms: $\lim _{k\to \infty }{\frac {1}{k}}\;=\;0$ Nevertheless, the harmonic series diverges.\, It is seen if we first group the terms with parentheses: $1+{\frac {1}{2}}+\left({\frac {1}{3}}+{\frac {1}{4}}\right)+\left({\frac {1}{5}}+{\frac {1}{6}}+{\frac {1}{7}}+{\frac {1}{8}}\right)+\left({\frac {1}{9}}+{\frac {1}{10}}+\ldots +{\frac {1}{16}}\right)+\ldots$ Here, each parenthetic sum contains a number of terms twice as many as the preceding one.\, The sum in the first parentheses is greater than\, $2\cdot {\frac {1}{4}}={\frac {1}{2}}$,\, the sum in the second parentheses is greater than\, $4\cdot {\frac {1}{8}}={\frac {1}{2}}$;\, thus one sees that the sum in all parentheses is greater than ${\frac {1}{2}}$.\, Consequently, the partial sum of $n$ first terms exceeds any given real number, when $n$ is sufficiently big.\\
The divergence of the harmonic series is very slow, though.\, Its speed may be illustrated by considering the difference $\sum _{k=1}^{n-1}{\frac {1}{k}}-\!\int _{1}^{n}{\frac {dx}{x}}\;=\;\sum _{k=1}^{n-1}{\frac {1}{k}}-\ln {n}$ (see the diagram).\, We know that $\ln {n}$ increases very slowly as $n\to \infty$ (e.g. $\ln {1\,000\,000\,000}\,\approx \,20.7$).\, The increasing of the partial sum $\sum _{k=1}^{n-1}{\frac {1}{k}}$ is about the same, since the limit $\lim _{n\to \infty }\left(\sum _{k=1}^{n-1}{\frac {1}{k}}-\ln {n}\right)\;=\;\gamma$ is a little positive number $\gamma \;=\;0.5772156649...$ which is called the Euler constant or Euler--Mascheroni constant .
   \begin{pspicture}(-0.5,-0.5)(6,6) \psaxes[Dx=1,Dy=1]{->}(0,0)(-0.5,-0.5)(5.5,5.5) \rput(5.5,-0.2){$x$} \rput(-0.2,5.5){$y$} \rput(-0.3,-0.3){$0$} \psplot[linecolor=red]{0.22}{5.2}{1 x div} \psline(1,0)(1,1) \psline[linecolor=blue](1,1)(2,1)(2,0.5)(3,0.5)(3,0.333)(4,0.333)(4,0.25)(5,0.25)(5,0.2) \psline(2,0)(2,0.5) \psline(3,0)(3,0.333) \psline(4,0)(4,0.25) \rput(1,3.5){$y={\frac {1}{x}}$} \end{pspicture}
# PlanetPhysics/Harmonic Series Diagram Html Rendering Test

The following is the missing graphics for the harmonic series diagram--as well as the limit of the harmonic series-- currently visible only in the page images mode only.
   \begin{pspicture}(-0.5,-0.5)(6,6) \psaxes[Dx=1,Dy=1]{->}(0,0)(-0.5,-0.5)(5.5,5.5) \rput(5.5,-0.2){$x$} \rput(-0.2,5.5){$y$} \rput(-0.3,-0.3){$0$} \psplot[linecolor=red]{0.22}{5.2}{1 x div} \psline(1,0)(1,1) \psline[linecolor=blue](1,1)(2,1)(2,0.5)(3,0.5)(3,0.333)(4,0.333)(4,0.25)(5,0.25)(5,0.2) \psline(2,0)(2,0.5) \psline(3,0)(3,0.333) \psline(4,0)(4,0.25) \rput(1,3.5){$y={\frac {1}{x}}$} \end{pspicture}
$\lim _{n\to \infty }\left(\sum _{k=1}^{n-1}{\frac {1}{k}}-\ln {n}\right)\;=\;\gamma$
# PlanetPhysics/Hamiltonian Algebroid 2
### Introduction

Hamiltonian algebroids are generalizations of the Lie algebras of canonical transformations.
Let $X$ and $Y$ be two vector fields on a smooth manifold $M$, represented here as operators acting on functions. Their commutator, or Lie bracket, $L$, is :
Failed to parse (unknown function "\begin{matrix}"): {\displaystyle \begin{matrix} [X,Y](f)=X(Y(f))-Y(X(f)). \end {align*} Moreover, consider the classical configuration space <math>Q = \bR^3} of a classical, mechanical system, or particle whose phase space is the cotangent bundle Failed to parse (unknown function "\bR"): {\displaystyle T^* \bR^3 \cong \bR^6} , for which the space of (classical) observables is taken to be the real vector space of smooth functions on $M$, and with T being an element of a Jordan-Lie (Poisson) algebra whose definition is also recalled next. Thus, one defines as in classical dynamics the Poisson algebra as a Jordan algebra in which $\circ$ is associative. We recall that one needs to consider first a specific algebra (defined as a vector space $E$ over a ground field (typically Failed to parse (unknown function "\bR"): {\displaystyle \bR} or Failed to parse (unknown function "\bC"): {\displaystyle \bC} )) equipped with a bilinear and distributive multiplication $\circ$~. Then one defines a Jordan algebra (over Failed to parse (unknown function "\bR"): {\displaystyle \bR} ), as a a specific algebra over Failed to parse (unknown function "\bR"): {\displaystyle \bR} for which:
</math> S \circ T &= T \circ S~, \\ S \circ (T \circ S^2) &= (S \circ T) \circ S^2 , ,$forallelements<math>S,T$ of this algebra.
Then, the usual algebraic types of morphisms automorphism, isomorphism, etc.) apply to a Jordan-Lie (Poisson) algebra defined as a real vector space Failed to parse (unknown function "\bR"): {\displaystyle \mathfrak A_{\bR}} together with a Jordan product $\circ$ and Poisson bracket
$\{~,~\}$, satisfying~:
Thus, the canonical transformations of the Poisson sigma model phase space specified by the Jordan-Lie (Poisson) algebra (also Poisson algebra), which is determined by both the Poisson bracket and the Jordan product $\circ$, define a Hamiltonian algebroid with the Lie brackets $L$ related to such a Poisson structure on the target space.
# PlanetPhysics/Hamiltonian Operator 3

Definition 0.1 The {Hamiltonian operator} H introduced in quantum mechanics (QM) by Schr\"odinger (and thus sometimes also called the Schr\"odinger operator) on the Hilbert space Failed to parse (unknown function "\Rset"): {\displaystyle L^2(\Rset^n)} is given by the action:
   Failed to parse (unknown function "\Rset"): {\displaystyle \psi \mapsto [-\nabla^2 +V(x)]\psi, \quad\psi\in L^2(\Rset^n), }
The operator defined above $[-\nabla ^{2}+V(x)]$ , for a potential function $V(x)$ specified as the real-valued function Failed to parse (unknown function "\Rset"): {\displaystyle V\colon \Rset^n \to \Rset} is called the Hamiltonian operator , H , and only very rarely the Schr\"odinger operator .

### Schr\"odinger formulation of QM

The energy conservation (quantum) law written with the operator H as the Schr\"odinger equation is fundamental in quantum mechanics and is perhaps the most utilized, mathematical computation device in quantum mechanics of systems with a finite number of degrees of freedom. There is also, however, the alternative approach in the Heisenberg picture, or formulation, in which the observable and other operators are time-dependent whereas the state vectors $\psi$ are time-independent, which reverses the time dependences betwen operators and state vectors from the more popular Schr\"odinger formulation. Other formulations of quantum theories occur in quantum field theories (QFT), such as QED (quantum electrodynamics) and QCD (quantum chromodynamics).

### Heisenberg formulation of QM

Although the two formulations, or pictures, are unitarily (or mathematically) equivalent, however, sometimes the claim is made that the Heisenberg picture is "more natural and fundamental than the Schr\"odinger" formulation because the Lorentz invariance from general relativity is also encountered in the Heisenberg picture, and also because there is a `correspondence' between the commutator of an observable operator with the Hamiltonian operator, and the Poisson bracket formulation of classical mechanics. If the state vector $\psi$, or $\left|\psi \right\rangle$ does not change with time as in the Heisenberg picture, then the `equation of motion' of a (quantum) observable operator is :
   ${\frac {d}{dt}}A_{quantum}=(i\hbar )^{-1}[A,H]+\left({\frac {\partial A}{\partial t}}\right)_{classical}$
# PlanetPhysics/Borel Isomorphism

A Borel isomorphism between two Borel spaces $(X;\mathbb {B} (X))$ and $(Y;\mathbb {B} (Y))$ is defined as the bijection $\iota :(X;\mathbb {B} (X))\cong (Y;\mathbb {B} (Y))$.
# PlanetPhysics/Bose-Einstein Condensation
## Bose--Einstein Condensation (BEC)

is described as the process occurring only very close to absolute zero (for example, below 1 mK) in which a large fraction of a dilute gas of weakly interacting bosons confined in an external potential have a lowest quantum state (or states) in which their wave functions overlap; hence the name of Bose--Einstein condensate (BEC) attributed to such a system of supercooled/ultracold, weakly interacting bosons with overlapping wave functions.
The original theoretical paper was published in 1925 in \emph{Zeitschrift fur Physik} by Satyendra Nath Bose in the German translation of Albert Einstein, and followed in a few theoretical papers by Einstein himself. Recently, there have been several experimental claims reported to have observed BECs in alkali metal gases at very low temperatures (see the following selected bibliography).
For a general introduction to BECs there is also available a related paper in PDF format.

### Bose-Einstein condensation in the alkali gases

A fundamental theoretical paper on this subject is: "Bose-Einstein condensation in the alkali gases: Some fundamental concepts" [Rev. Mod. Phys. 73, 307 (2001)], by Anthony J. Leggett.
# PlanetPhysics/Boson

Bosons are particles with integer spin, named after the Indian physicist Satyendra Nath Bose. They are described by Bose-Einstein statistics, meaning that two bosons can occupy the same quantum state.

### List of bosons

* Photon W boson Z boson Higgs boson graviton gluon
# PlanetPhysics/C2 Category

\newcommand{\sqdiagram}[9]{Failed to parse (unknown function "\diagram"): {\displaystyle \diagram #1 \rto^{#2} \dto_{#4}& \eqno{\mbox{#9}}} }
A $C_{2}$-category is defined as a category ${\mathcal {C}}$ that has products, coproducts and a zero object, and if the morphism $\iota :\oplus A_{i}\to \mathbf {X} A_{i}$ is a monomorphism for any family of objects $\left\{A_{i}\right\}$ in ${\mathcal {C}}$ (p. 81 in).
One readily obtains the result that a $C_{2}$-category is $C_{1}$.
# PlanetPhysics/CCG
## Class of continuous functions on a topological groupoid with compact support

Def. $C_{c}({\mathsf {G}})$ is defined as the class of continuous functions acting on a topological groupoid ${\mathsf {G}}$ with compact support, and with values in a normed space, field, or algebraic structure. In most applications it will, however, suffice to select ${\mathsf {G}}$ as a locally compact (topological) groupoid, ${\mathsf {G}}_{lc}$, support for this space of continuous functions.
Note: this concept is important for quantum theoretical applications related to quantum groupoids defined by a $C_{c}({\mathsf {G}})$.
# PlanetPhysics/CCR Representation Theory

In connection with the Schr\"odinger representation, one defines a Schr\"odinger d-system as a set $\left\{Q_{j},P_{j}\right\}_{j=1}^{d}$ of self-adjoint operators on a Hilbert space ${\mathcal {H}}$ (such as the position and momentum operators, for example) when there exist mutually orthogonal closed subspaces ${\mathcal {H}}_{\alpha }$ of ${\mathcal {H}}$ such that ${\mathcal {H}}=\oplus _{\alpha }{\mathcal {H}}_{\alpha }$ with the following two properties:
* (i) each ${\mathcal {H}}_{\alpha }$ reduces all $Q_{j}$ and all $P_{j}$ ;
* (ii) the set $\left\{Q_{j},P_{j}\right\}_{j=1}^{d}$ is, in each ${\mathcal {H}}_{\alpha }$, unitarily equivalent to the Schr\"odinger representation $\left\{Q_{j}^{S},P_{j}^{S}\right\}_{j=1}^{d},$.
A set $\left\{Q_{j},P_{j}\right\}_{j=1}^{d}$ of self-adjoint operators on a Hilbert space ${\mathcal {H}}$ is called a Weyl representation with $d$ degrees of freedom if $Q_{j}$ and $P_{j}$ satisfy the Weyl relations:
* Failed to parse (syntax error): {\displaystyle e^{itQ_j} \dot e^{isP_k} = e^{âˆ’ist} \hbar_{jk} e^{isP_k} \dot e^{itQ_j},}
* $e^{itQ_{j}}{\dot {e}}^{isQ_{k}}=e^{isQ_{k}}{\dot {e}}^{itQ_{j}},$
* $e^{itP_{j}}{\dot {e}}{isP_{k}}=e^{isP_{k}}{\dot {e}}^{itP_{j}},$
with $j,k=1,...,d,s,t\in \mathbb {R}$.
The Schr\"odinger representation $\left\{Q_{j},P_{j}\right\}_{j=1}^{d}$ is a Weyl representation of CCR.
Von Neumann established a uniqueness \htmladdnormallink{theorem {http://planetphysics.us/encyclopedia/Formula.html}: if the Hilbert space ${\mathcal {H}}$ is separable, then every Weyl representation of CCR with $d$ degrees of freedom is a Schr\"odinger $d$-system}. Since the pioneering work of von Neumann there have been numerous reports published concerning representation theory of CCR (viz. ref. and references cited therein).
# PlanetPhysics/CW Complex Representation Theorems
### CW-complex representation theorems in quantum algebraic topology

\htmladdnormallink{QAT {http://planetphysics.us/encyclopedia/QAT.html} theorems for quantum state spaces of spin networks and quantum spin foams based on $CW$, $n$-connected models and fundamental theorems.}
Let us consider first a lemma in order to facilitate the proof of the following theorem concerning spin networks and quantum spin foams.
Lemma Let $Z$ be a $CW$ complex that has the (three--dimensional) Quantum Spin `Foam' (QSF) as a subspace. Furthermore, let $f:Z\rightarrow QSS$ be a map so that Failed to parse (syntax error): {\displaystyle f \mid QSF = 1_{QSF'' } , with QSS being an arbitrary, local quantum state space (which is not necessarily finite). There exists an $n$-connected $CW$ model (Z,QSF) for the pair (QSS,QSF) such that}:
$f_{*}:\pi _{i}(Z)\rightarrow \pi _{i}(QST)$,
is an isomorphism for $i>n$ and it is a monomorphism for $i=n$. The $n$-connected $CW$ model is unique up to homotopy equivalence. (The $CW$ complex, $Z$, considered here is a homotopic `hybrid' between QSF and QSS).
Theorem 2. (Baianu, Brown and Glazebrook, 2007: In Section 9 of a recent NAQAT preprint). For every pair $(QSS,QSF)$ of topological spaces defined as in Lemma 1 , with QSF nonempty, there exist $n$-connected $CW$ models $f:(Z,QSF)\rightarrow (QSS,QSF)$ for all $n\geq 0$. Such models can be then selected to have the property that the $CW$ complex $Z$ is obtained from QSF by attaching cells of dimension $n>2$, and therefore $(Z,QSF)$ is $n$-connected. Following Lemma 01 one also has that the map: $f_{*}:\pi _{i}(Z)\rightarrow \pi _{i}(QSS)$ which is an isomorphism for $i>n$, and it is a monomorphism for $i=n$.
Note See also the definitions of (quantum) \htmladdnormallink{spin networks and spin foams {http://planetphysics.us/encyclopedia/SpinNetworksAndSpinFoams.html}.}
# PlanetPhysics/Borel Groupoid
### Definitions

* Borel function A function $f_{B}:(X;{\mathcal {B}})\to (X;{\mathcal {C}}$) of Borel spaces is defined to be a Borel function if the inverse image of every Borel set under $f_{B}^{-1}$ is also a Borel set.
* Borel groupoid Let Failed to parse (unknown function "\grp"): {\displaystyle \grp} be a groupoid and Failed to parse (unknown function "\grp"): {\displaystyle \grp^{(2)}} a subset of Failed to parse (unknown function "\grp"): {\displaystyle \grp \times \grp} -- the set of its composable pairs. A Borel groupoid is defined as a groupoid Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\displaystyle \grp_B} such that Failed to parse (unknown function "\grp"): {\displaystyle \grp_B^{(2)}} is a Borel set in the product structure on Failed to parse (unknown function "\grp"): {\displaystyle \grp_B \times \grp_B} , and also such that the functions $(x,y)\mapsto xy$ from Failed to parse (unknown function "\grp"): {\displaystyle \grp_B^{(2)}} to Failed to parse (unknown function "\grp"): {\displaystyle \grp_B} , and $x\mapsto x^{-1}$ from Failed to parse (unknown function "\grp"): {\displaystyle \grp_B} to Failed to parse (unknown function "\grp"): {\displaystyle \grp_B} are all (measurable) Borel functions (ref.).

#### Analytic Borel space

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\displaystyle \grp_B} becomes an analytic groupoid if its Borel structure is analytic.
A Borel space $(X;{\mathcal {B}})$ is called analytic if it is countably separated, and also if it is the image of a Borel function from a standard Borel space.
# PlanetPhysics/Borel Morphism

Let Failed to parse (unknown function "\grp"): {\displaystyle \grp_B} and Failed to parse (unknown function "\grp"): {\displaystyle \grp_B} * be two groupoids whose object spaces are Borel. An \htmladdnormallink{algebraic {http://planetphysics.us/encyclopedia/CoIntersections.html} morphism} from Failed to parse (unknown function "\grp"): {\displaystyle \grp_B}
to Failed to parse (unknown function "\grp"): {\displaystyle \grp_B} * is defined as a left action of Failed to parse (unknown function "\grp"): {\displaystyle \grp_B} on Failed to parse (unknown function "\grp"): {\displaystyle \grp_B} * which commutes with the multiplication on Failed to parse (unknown function "\grp"): {\displaystyle \grp_B} . Such an algebraic morphism between Borel groupoids is said to be a Borel morphism if the action of Failed to parse (unknown function "\grp"): {\displaystyle \grp_B} on Failed to parse (unknown function "\grp"): {\displaystyle \grp_B} * is Borel (viz. ref.)
# PlanetPhysics/Borel Space

A Borel space $(X;{\mathcal {B}}(X))$ is defined as a set $X$, together with a Borel $\sigma$-algebra ${\mathcal {B}}(X)$ of subsets of $X$, called Borel sets. The Borel algebra on $X$ is the smallest $\sigma$-algebra containing all open sets (or, equivalently, all closed sets if the topology on closed sets is selected).
Borel sets were named after the French mathematician Emile Borel.
A subspace of a Borel space $(X;{\mathcal {B}}(X))$ is a subset $S\subset X$ endowed with the relative Borel structure, that is the $\sigma$-algebra of all subsets of $S$ of the form $S\bigcap E$, where $E$ is a Borel subset of $X$.
A rigid Borel space $(X_{r};{\mathcal {B}}(X_{r}))$ is defined as a Borel space whose only automorphism $f:X_{r}\to X_{r}$ (that is, with $f$ being a bijection, and also with $f(A)=f^{-1}(A)$ for any $A\in {\mathcal {B}}(X_{r})$) is the identity function $1_{(X_{r};{\mathcal {B}}(X_{r}))}$ (ref.).
R. M. Shortt and J. Van Mill provided the first construction of a rigid Borel space on a `set of large cardinality'.
# Real numbers/Series/Cauchy-criterion/Fact/Proof

We set $x_{n}:=\sum _{k=0}^{n}a_{k}$. The convergence of the series means the convergence of this sequence of partial sums. A real sequence converges if and only if it is a Cauchy-sequence. This means that for every $\epsilon >0$ there exists some $n_{0}$ such that for all
    $n\geq m\geq n_{0}\,$
the estimate
    $\vert {x_{n}-x_{m}}\vert \leq \epsilon \,$
holds. In the case of a series this just means
    $\vert {x_{n}-x_{m}}\vert =\vert {\sum _{k=0}^{n}a_{k}-\sum _{k=0}^{m}a_{k}}\vert =\vert {\sum _{k=m+1}^{n}a_{k}}\vert \leq \epsilon \,.$
# Real numbers/Sequence/Tending to infinity/Definition

$+\infty$
A real sequence ${\left(x_{n}\right)}_{n\in \mathbb {N} }$ is said to tend to $+\infty$, if for every $s\in \mathbb {R}$, there exists some $N\in \mathbb {N}$, such that
    $x_{n}\geq s{\text{ holds for all }}n\geq N.$
The sequence is said to tend to $-\infty$, if for every $s\in \mathbb {R}$. there exists some $N\in \mathbb {N}$. such that
    $x_{n}\leq s{\text{ holds for all }}n\geq N.$
# Real numbers/Sequence/Unique limit/Fact

A real sequence
# Real numbers/Sequences/Squeeze criterion/Fact

Let ${\left(x_{n}\right)}_{n\in \mathbb {N} },\,{\left(y_{n}\right)}_{n\in \mathbb {N} }$ and ${\left(z_{n}\right)}_{n\in \mathbb {N} }$ denote real sequences. Suppose that
    $x_{n}\leq y_{n}\leq z_{n}{\text{ for all }}n\in \mathbb {N}$
and that ${\left(x_{n}\right)}_{n\in \mathbb {N} }$ and ${\left(z_{n}\right)}_{n\in \mathbb {N} }$ converge to the same limit $a$.
${\left(y_{n}\right)}_{n\in \mathbb {N} }$
$a$
# Real numbers/Series/Cauchy-criterion/Fact/Proof/Exercise

Prove the Cauchy criterion for series of real numbers.
# Real numbers/Sequence/Limit and convergence/Definition

Let ${\left(x_{n}\right)}_{n\in \mathbb {N} }$ denote a real sequence, and let $x\in \mathbb {R}$. We say that the sequence converges to $x$, if the following property holds.
For every positive $\epsilon >0$, $\epsilon \in \mathbb {R}$, there exists some $n_{0}\in \mathbb {N}$, such that for all $n\geq n_{0}$, the estimate
    $\vert {x_{n}-x}\vert \leq \epsilon \,$
holds.
If this condition is fulfilled, then $x$ is called the limit of the sequence. For this we write
    $\lim _{n\rightarrow \infty }x_{n}:=x\,.$
If the sequence converges to a limit, we just say that the sequence converges, otherwise, that the sequence diverges.
# Real numbers/Sequence/Unique limit/Fact/Proof

We assume that the sequence has two distinct limits $x,y$, $x\neq y$. Then $d:=\vert {x-y}\vert >0$. We consider $\epsilon :=d/3>0$. Because of the convergence to $x$ there exists an $n_{0}$ such that
    $\vert {x_{n}-x}\vert \leq \epsilon {\text{ for all }}n\geq n_{0}$
and because of the convergence to $y$ there exists an $n_{0}'$ such that
    $\vert {x_{n}-y}\vert \leq \epsilon {\text{ for all }}n\geq n_{0}'.$
hence both conditions hold simultaneously for $n\geq \max\{n_{0},n_{0}'\}$. Suppose that $n$ is as large as this maximum. Then due to the triangle inequality we arrive at the contradiction
    $d=\vert {x-y}\vert \leq \vert {x-x_{n}}\vert +\vert {x_{n}-y}\vert \leq \epsilon +\epsilon =2d/3\,.$
# Real numbers/Sequences/Bounded increasing/Convergence/Decimal expansion/Section
## Corollary

A bounded and monotone sequence
$\mathbb {R}$

### Proof

Due to the condition, the sequence is increasing and bounded from above or decreasing and bounded from below. Because of fact, we have a Cauchy sequence which converges in $\mathbb {R}$.
$\Box$
This statement is also the reason that any decimal expansion defines a real number. An (infinite) decimal expansion
    $a.a_{-1}a_{-2}a_{-3}\ldots$
with $a\in \mathbb {N}$ (we restrict to nonnegative numbers) and $a_{-n}\in \{0,\ldots ,9\}$ is just the sequence of rational numbers
    $x_{0}:=a,\,x_{1}:=a+a_{-1}\cdot {\frac {1}{10}},\,x_{2}:=a+a_{-1}\cdot {\frac {1}{10}}+a_{-2}\cdot {\left({\frac {1}{10}}\right)}^{2},\,{\rm {{etc}.}}$
This sequence is increasing. It is also bounded, e.g. by $a+1$, so that it defines a Cauchy sequence and thus a real number.
# Real numbers/Series/Cauchy-criterion/Fact

Let
    $\sum _{k=0}^{\infty }a_{k}$
be a series of real numbers.
convergent if and only if the following Cauchy-criterion holds: For every $\epsilon >0$ there exists some $n_{0}$ such that for all
    $n\geq m\geq n_{0}\,$
the estimate
    $\vert {\sum _{k=m}^{n}a_{k}}\vert \leq \epsilon \,$
# Real numbers/Sequences/Squeeze criterion/Fact/Proof/Exercise

Let ${\left(x_{n}\right)}_{n\in \mathbb {N} },\,{\left(y_{n}\right)}_{n\in \mathbb {N} }$ and ${\left(z_{n}\right)}_{n\in \mathbb {N} }$ be three real sequences. Let $x_{n}\leq y_{n}\leq z_{n}$ for all $n\in \mathbb {N}$ and ${\left(x_{n}\right)}_{n\in \mathbb {N} }$ and ${\left(z_{n}\right)}_{n\in \mathbb {N} }$ be convergent to the same limit $a$. Prove that also ${\left(y_{n}\right)}_{n\in \mathbb {N} }$ converges to the same limit $a$.