Proof of $\mathbb{R}^n$ with the Euclidean Distance Is a Complete Metric Space with Coordinatewise Convergence

Each result cited is universally quantified over the data in its own statement.

The rules of arithmetic and order in the ordered field $\mathbb{R}$, carried by Commutative Rings, Fields and Ordered Fields: Standard Notation §ordered-fields, are used without further citation. For $u,v\in\mathbb{R}^{n}$ and $t\in\mathbb{R}$, $u+v$, $u-v$, $-u$ and $tu$ lie in $\mathbb{R}^{n}$ and have the components given by Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations §operations; elements of $\mathbb{R}^{n}$ with the same components are equal by Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations §equality, and the origin $0$ has all components $0$ by Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations §origin and Pointwise Operations on Real-Valued Maps §constants. By The Euclidean Distance on $\mathbb{R}^n$ §distance, $d_{E}(u,v)=\lVert u-v\rVert$. By Metric Spaces: Balls, the Metric Topology, and the Real and Complex Numbers as Metric Spaces §real-line, The Distance on the Real Numbers §distance and The Distance on the Real Numbers Is a Map with Values the Absolute Differences §map, $d_{\mathbb{R}}$ is a metric on $\mathbb{R}$ with $d_{\mathbb{R}}(a,b)=|a-b|$.

Coordinate bounds. For $u,v\in\mathbb{R}^{n}$, $k\in[n]$ and $s\in\mathbb{R}$, applying Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square §coordinate to $u-v$, whose components are $u_{j}-v_{j}$, gives

$$|u_{k}-v_{k}|\le d_{E}(u,v),\qquad\text{and}\qquad d_{E}(u,v)\le n\,s\ \text{ if }|u_{j}-v_{j}|\le s\text{ for every }j\in[n].\tag{B}$$

Clause metric. By The Euclidean Distance on $\mathbb{R}^n$ §distance and The Summation Map on Real-Valued Maps over a Finite Set, and the Dot Product, Euclidean Norm and Euclidean Distance as Maps §distance, $d_{E}$ is a map $\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}$. Let $x,y,z\in\mathbb{R}^{n}$. Nonnegative: $d_{E}(x,y)=\lVert x-y\rVert\ge0$ by Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square §square. Zero: by Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square §vanishing, $d_{E}(x,y)=0$ if and only if $x-y=0$; comparing components, this holds if and only if $x_{k}-y_{k}=0$, that is $x_{k}=y_{k}$, for every $k\in[n]$, that is, if and only if $x=y$. Symmetric: $y-x=-(x-y)$, since $y_{k}-x_{k}=-(x_{k}-y_{k})$ for every $k\in[n]$; so $d_{E}(y,x)=\lVert-(x-y)\rVert=\lVert x-y\rVert=d_{E}(x,y)$ by Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square §homogeneity. Triangle: $x-z=(x-y)+(y-z)$, since $x_{k}-z_{k}=(x_{k}-y_{k})+(y_{k}-z_{k})$ for every $k\in[n]$; so $d_{E}(x,z)\le\lVert x-y\rVert+\lVert y-z\rVert=d_{E}(x,y)+d_{E}(y,z)$ by Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square §triangle. Hence $d_{E}$ is a metric on $\mathbb{R}^{n}$ in the sense of Metric Space §metric, and $(\mathbb{R}^{n},d_{E})$ is a metric space by Metric Space §metric-space.

For a sequence $(x^{(m)})$ in $\mathbb{R}^{n}$, that is a map $\mathbb{N}\to\mathbb{R}^{n}$ by Sequences §sequence, and $k\in[n]$, the coordinate sequence $m\mapsto x^{(m)}_{k}$ is a map $\mathbb{N}\to\mathbb{R}$ by Sets and Maps: Ordinary Notation §maps, its condition being predicative by Predicative Substitution of Values of Functions and Binary Operations, and Predicative Formulas for Negatives, Differences, Absolute Values, Distances, Reciprocals and Quotients in an Ordered Field §value-substitution, hence a sequence in $\mathbb{R}$.

A finite maximum. We use the following twice. Let, for each $k\in[n]$, $S_{k}=\{N'\in\mathbb{N}:P_{k}(N')\}$ be a nonempty subset of $\mathbb{N}$, formed by Sets and Maps: Ordinary Notation §set-builder, where $P_{k}(N')$ is a property expressed by a formula in $k$ and $N'$. The order of $\mathbb{N}_{0}$ is a well-order by The Natural Numbers and the Natural Numbers with Zero: Arithmetic, Order, Induction and Recursion §order, hence a total order by Well-Orders on a Set §well-order, and $S_{k}$ has a least element; it is unique by Bounds, Least and Greatest Elements, Suprema and Infima for a Partial Order §least, so $N_{k}=\min S_{k}$ is defined, and $k\mapsto N_{k}$ is a map $[n]\to\mathbb{N}$ by Sets and Maps: Ordinary Notation §maps. No choice is involved: each $N_{k}$ is the least element of a definite set. The interval $[n]=\{1,\dots,n\}$ (Intervals of Natural Numbers §segment) is finite by Images of Finite Sets Are Finite, Bounded Sets of Natural Numbers Are Finite, and Nonempty Finite Totally Ordered Sets Have Extreme Elements §naturals, so its image $T=\{N_{k}:k\in[n]\}$ is finite by Images of Finite Sets Are Finite, Bounded Sets of Natural Numbers Are Finite, and Nonempty Finite Totally Ordered Sets Have Extreme Elements §image; $T$ is nonempty because $1\in[n]$ (as $1\le n$ by The Natural Numbers and the Natural Numbers with Zero: Arithmetic, Order, Induction and Recursion §order), and $T\subseteq\mathbb{N}\subseteq\mathbb{N}_{0}$. By Images of Finite Sets Are Finite, Bounded Sets of Natural Numbers Are Finite, and Nonempty Finite Totally Ordered Sets Have Extreme Elements §extremes, $T$ has a greatest element $N=\max T$, unique by Bounds, Least and Greatest Elements, Suprema and Infima for a Partial Order §least; then $N\in\mathbb{N}$ and $N_{k}\le N$ for every $k\in[n]$.

Also $n\ge1$ in $\mathbb{N}_{0}$, hence $n\ge1>0$ in $\mathbb{R}$ by The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness §agreement, where $n\in\mathbb{R}$ by Closure Facts in an Ordered Field: Negatives, Differences, Quotients, Absolute Values, Powers, Natural Numbers, Halves and Minima §naturals. For a real number $\varepsilon>0$ put $\delta=\varepsilon/(2n)$; then $\delta>0$ and $n\,\delta=\varepsilon/2<\varepsilon$. Indeed, $2=1+1$ in $\mathbb{R}$ by Commutative Rings, Fields and Ordered Fields: Standard Notation §numerals and Closure Facts in an Ordered Field: Negatives, Differences, Quotients, Absolute Values, Powers, Natural Numbers, Halves and Minima §two, and $0<1+1$ by Rules of Arithmetic and Order in an Ordered Field §midpoint, so $2n>0$ since $2>0$ and $n>0$; hence $(2n)^{-1}>0$ by Rules of Arithmetic and Order in an Ordered Field §positive-reciprocal, and $\delta=\varepsilon\,(2n)^{-1}>0$, where $(2n)^{-1}$ and $\delta$ lie in $\mathbb{R}$ by Closure Facts in an Ordered Field: Negatives, Differences, Quotients, Absolute Values, Powers, Natural Numbers, Halves and Minima §quotient, as $2n\neq0$. Moreover $n\,\delta=\varepsilon/2=(0+\varepsilon)/(1+1)$, with $\varepsilon/2\in\mathbb{R}$ by Closure Facts in an Ordered Field: Negatives, Differences, Quotients, Absolute Values, Powers, Natural Numbers, Halves and Minima §halves, and $0<\varepsilon/2<\varepsilon$ by Rules of Arithmetic and Order in an Ordered Field §midpoint with $x=0$ and $y=\varepsilon$.

Clause coordinatewise. Let $(x^{(m)})$ be a sequence in $\mathbb{R}^{n}$ and $x\in\mathbb{R}^{n}$.

Suppose $(x^{(m)})$ converges to $x$ in $(\mathbb{R}^{n},d_{E})$, and let $k\in[n]$ and $\varepsilon>0$. By Convergent Sequence in a Metric Space §converges there is $N\in\mathbb{N}$ with $d_{E}(x^{(m)},x)<\varepsilon$ for every $m\ge N$; for these $m$, $d_{\mathbb{R}}(x^{(m)}_{k},x_{k})=|x^{(m)}_{k}-x_{k}|\le d_{E}(x^{(m)},x)<\varepsilon$ by (B). So $(x^{(m)}_{k})$ converges to $x_{k}$ in $(\mathbb{R},d_{\mathbb{R}})$ by Convergent Sequence in a Metric Space §converges.

Conversely suppose $(x^{(m)}_{k})$ converges to $x_{k}$ in $(\mathbb{R},d_{\mathbb{R}})$ for every $k\in[n]$. Let $\varepsilon>0$; first fix $\delta=\varepsilon/(2n)$ as above. For $k\in[n]$ let $S_{k}$ be the set of $N'\in\mathbb{N}$ such that $|x^{(m)}_{k}-x_{k}|<\delta$ for every $m\in\mathbb{N}$ with $m\ge N'$, a condition that is predicative by Predicative Substitution of Values of Functions and Binary Operations, and Predicative Formulas for Negatives, Differences, Absolute Values, Distances, Reciprocals and Quotients in an Ordered Field §value-substitution and Predicative Substitution of Values of Functions and Binary Operations, and Predicative Formulas for Negatives, Differences, Absolute Values, Distances, Reciprocals and Quotients in an Ordered Field §distance-formula; it is nonempty by Convergent Sequence in a Metric Space §converges applied with $\delta$ to the $k$-th coordinate sequence. Next let $N_{k}=\min S_{k}$ and $N=\max\{N_{k}:k\in[n]\}$ as in the paragraph on a finite maximum. For $m\ge N$ and $k\in[n]$ we have $m\ge N_{k}$, so $|x^{(m)}_{k}-x_{k}|<\delta$; by (B) with $s=\delta$, $d_{E}(x^{(m)},x)\le n\,\delta<\varepsilon$. Hence $(x^{(m)})$ converges to $x$ in $(\mathbb{R}^{n},d_{E})$ by Convergent Sequence in a Metric Space §converges.

Clause cauchy. Let $(x^{(m)})$ be a sequence in $\mathbb{R}^{n}$.

Suppose it is Cauchy in $(\mathbb{R}^{n},d_{E})$, and let $k\in[n]$ and $\varepsilon>0$. By Cauchy Sequence in a Metric Space §cauchy there is $N\in\mathbb{N}$ with $d_{E}(x^{(m)},x^{(\ell)})<\varepsilon$ for all $m,\ell\ge N$, and then $d_{\mathbb{R}}(x^{(m)}_{k},x^{(\ell)}_{k})=|x^{(m)}_{k}-x^{(\ell)}_{k}|<\varepsilon$ by (B). So $(x^{(m)}_{k})$ is Cauchy in $(\mathbb{R},d_{\mathbb{R}})$.

Conversely suppose $(x^{(m)}_{k})$ is Cauchy in $(\mathbb{R},d_{\mathbb{R}})$ for every $k\in[n]$. Let $\varepsilon>0$; first fix $\delta=\varepsilon/(2n)$. For $k\in[n]$ let $S_{k}$ be the set of $N'\in\mathbb{N}$ such that $|x^{(m)}_{k}-x^{(\ell)}_{k}|<\delta$ for all $m,\ell\in\mathbb{N}$ with $m\ge N'$ and $\ell\ge N'$, a condition that is predicative by Predicative Substitution of Values of Functions and Binary Operations, and Predicative Formulas for Negatives, Differences, Absolute Values, Distances, Reciprocals and Quotients in an Ordered Field §value-substitution and Predicative Substitution of Values of Functions and Binary Operations, and Predicative Formulas for Negatives, Differences, Absolute Values, Distances, Reciprocals and Quotients in an Ordered Field §distance-formula; it is nonempty by Cauchy Sequence in a Metric Space §cauchy. Next let $N_{k}=\min S_{k}$ and $N=\max\{N_{k}:k\in[n]\}$ as above. For $m,\ell\ge N$ and every $k\in[n]$, $m,\ell\ge N_{k}$, so $|x^{(m)}_{k}-x^{(\ell)}_{k}|<\delta$, and by (B) $d_{E}(x^{(m)},x^{(\ell)})\le n\,\delta<\varepsilon$. Hence $(x^{(m)})$ is Cauchy in $(\mathbb{R}^{n},d_{E})$ by Cauchy Sequence in a Metric Space §cauchy.

Clause complete. Let $(x^{(m)})$ be a Cauchy sequence in $(\mathbb{R}^{n},d_{E})$. By the clause cauchy, for every $k\in[n]$ the sequence $(x^{(m)}_{k})$ is Cauchy in $(\mathbb{R},d_{\mathbb{R}})$, which is complete by Metric Spaces: Balls, the Metric Topology, and the Real and Complex Numbers as Metric Spaces §real-line; so it is convergent in $(\mathbb{R},d_{\mathbb{R}})$ by Complete Metric Space §complete, and it has the limit $\lim_{m\to\infty}x^{(m)}_{k}$, which exists and is unique by The Limit of a Convergent Sequence in a Metric Space §limit. Since $\lim_{m\to\infty}x^{(m)}_{k}\in\mathbb{R}$ for every $k\in[n]$, the map $L:[n]\to\mathbb{R}$, $k\mapsto\lim_{m\to\infty}x^{(m)}_{k}$, exists and is unique by Sets and Maps: Ordinary Notation §maps; $L\in\mathbb{R}^{n}$ by Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations §membership, with $L_{k}=\lim_{m\to\infty}x^{(m)}_{k}$. By The Limit of a Convergent Sequence in a Metric Space §converges-to-limit, $(x^{(m)}_{k})$ converges to $L_{k}$ in $(\mathbb{R},d_{\mathbb{R}})$ for every $k\in[n]$, so $(x^{(m)})$ converges to $L$ in $(\mathbb{R}^{n},d_{E})$ by the clause coordinatewise, and is convergent by Convergent Sequence in a Metric Space §convergent. Hence $(\mathbb{R}^{n},d_{E})$ is complete by Complete Metric Space §complete.

Clause lipschitz. Let $k\in[n]$. Since $x_{k}\in\mathbb{R}$ and $\lVert x\rVert\in\mathbb{R}$ for $x\in\mathbb{R}^{n}$, the maps $x\mapsto x_{k}$ and $x\mapsto\lVert x\rVert$ from $\mathbb{R}^{n}$ to $\mathbb{R}$ exist by Sets and Maps: Ordinary Notation §maps (the condition $z=x_{k}$ being predicative by Predicative Substitution of Values of Functions and Binary Operations, and Predicative Formulas for Negatives, Differences, Absolute Values, Distances, Reciprocals and Quotients in an Ordered Field §value-substitution) and by The Summation Map on Real-Valued Maps over a Finite Set, and the Dot Product, Euclidean Norm and Euclidean Distance as Maps §norm respectively. For $x,x'\in\mathbb{R}^{n}$, by (B),

$$d_{\mathbb{R}}(x_{k},x'_{k})=|x_{k}-x'_{k}|\le1\cdot d_{E}(x,x'),$$

and by Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square §reverse-triangle,

$$d_{\mathbb{R}}(\lVert x\rVert,\lVert x'\rVert)=\bigl|\lVert x\rVert-\lVert x'\rVert\bigr|\le\lVert x-x'\rVert=1\cdot d_{E}(x,x').$$

As $1\ge0$, both maps are Lipschitz with constant $1$ by Lipschitz Map Between Metric Spaces §lipschitz.

Clause limit-laws. Assume the hypotheses. By Sets and Maps: Ordinary Notation §maps, the defining conditions being predicative by Predicative Substitution of Values of Functions and Binary Operations, and Predicative Formulas for Negatives, Differences, Absolute Values, Distances, Reciprocals and Quotients in an Ordered Field §value-substitution and Predicative Substitution of Values of Functions and Binary Operations, and Predicative Formulas for Negatives, Differences, Absolute Values, Distances, Reciprocals and Quotients in an Ordered Field §operation-substitution (the dot product being a value of the map of The Summation Map on Real-Valued Maps over a Finite Set, and the Dot Product, Euclidean Norm and Euclidean Distance as Maps §dot-product), $m\mapsto x^{(m)}+y^{(m)}$ and $m\mapsto t_{m}x^{(m)}$ are sequences in $\mathbb{R}^{n}$ and $m\mapsto x^{(m)}\cdot y^{(m)}$ is a sequence in $\mathbb{R}$. Let $k\in[n]$. By the clause coordinatewise, $(x^{(m)}_{k})$ and $(y^{(m)}_{k})$ converge to $x_{k}$ and $y_{k}$ in $(\mathbb{R},d_{\mathbb{R}})$; by Convergence, Limits and the Cauchy Condition for Real Sequences Agree with Those in the Real Line as a Metric Space, which is Complete §converges, they, and $(t_{m})$, converge to $x_{k}$, $y_{k}$ and $t$ in the sense of Convergent Sequences of Real Numbers §converges. By Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences §arithmetic (sums and products),

$$(x^{(m)}+y^{(m)})_{k}=x^{(m)}_{k}+y^{(m)}_{k}\to x_{k}+y_{k}=(x+y)_{k},\qquad (t_{m}x^{(m)})_{k}=t_{m}\,x^{(m)}_{k}\to t\,x_{k}=(tx)_{k},$$

and $x^{(m)}_{k}y^{(m)}_{k}\to x_{k}y_{k}$, in the sense of Convergent Sequences of Real Numbers §converges; by Convergence, Limits and the Cauchy Condition for Real Sequences Agree with Those in the Real Line as a Metric Space, which is Complete §converges the first two convergences hold in $(\mathbb{R},d_{\mathbb{R}})$. As this holds for every $k\in[n]$, the clause coordinatewise gives that $(x^{(m)}+y^{(m)})$ converges to $x+y$ and $(t_{m}x^{(m)})$ converges to $tx$ in $(\mathbb{R}^{n},d_{E})$.

For the dot product, the maps $e:[n]\times\mathbb{N}\to\mathbb{R}$, $(k,m)\mapsto x^{(m)}_{k}y^{(m)}_{k}$, and $D:[n]\to\mathbb{R}$, $k\mapsto x_{k}y_{k}$, exist by Sets and Maps: Ordinary Notation §maps, their conditions being predicative by Predicative Substitution of Values of Functions and Binary Operations, and Predicative Formulas for Negatives, Differences, Absolute Values, Distances, Reciprocals and Quotients in an Ordered Field §value-substitution and Predicative Substitution of Values of Functions and Binary Operations, and Predicative Formulas for Negatives, Differences, Absolute Values, Distances, Reciprocals and Quotients in an Ordered Field §operation-substitution, and $(e(k,m))_{m\in\mathbb{N}}$ converges to $D_{k}$ for every $k$ in the finite set $[n]$ (finite by Images of Finite Sets Are Finite, Bounded Sets of Natural Numbers Are Finite, and Nonempty Finite Totally Ordered Sets Have Extreme Elements §naturals), as just shown. By Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences §finite-sums, $\bigl(\sum_{k\in[n]}e(k,m)\bigr)_{m\in\mathbb{N}}$ converges to $\sum_{k\in[n]}D_{k}$. Since $[n]=\{1,\dots,n\}$, these sums are $\sum_{k=1}^{n}x^{(m)}_{k}y^{(m)}_{k}=x^{(m)}\cdot y^{(m)}$ and $\sum_{k=1}^{n}x_{k}y_{k}=x\cdot y$, by Sums and Products over a Finite Set and over an Interval §intervals and The Dot Product on $\mathbb{R}^n$ §dot-product. So $(x^{(m)}\cdot y^{(m)})$ converges to $x\cdot y$, and by Convergence, Limits and the Cauchy Condition for Real Sequences Agree with Those in the Real Line as a Metric Space, which is Complete §converges it does so in $(\mathbb{R},d_{\mathbb{R}})$.