${\mathbb{R}}^n$ as the set of functions $n\to \mathbb{R}$, and $d_1$, $d_2$, $d_{\mathrm{\infty }}$ are metrics on it

Let $n\in \mathbb{N}$ with $n\ge 1$. A von Neumann natural is the set of its predecessors, $n=\{0,1,\dots ,n-1\}$ (The natural numbers $\mathbb{N}$ (von Neumann)), so it can be used directly as an index set. Define

${\mathbb{R}}^n:=\{x:x\text{ is a function }n\to \mathbb{R}\},$

and write $x_k$ for $x(k)$, $k<n$. Two elements of ${\mathbb{R}}^n$ are equal exactly when they agree at every $k<n$, functions being equal when they have the same values. For $x,y\in {\mathbb{R}}^n$ put

$d_1(x,y):={\sum }_{k<n}\mathrm{\mid }{x}_k-y_k\mathrm{\mid },d_2(x,y):=\sqrt{{\sum }_{k<n}(x_k-y_k{)}^2},d_{\mathrm{\infty }}(x,y):=\max \{\mathrm{\mid }{x}_k-y_k\mathrm{\mid }:k<n\}\mathrm{.}$

All three are well defined: the finite sums are those of Finite sums and finite products, by recursion; the sum of squares is nonnegative (Laws of finite sums and finite products, Squares of nonzero elements are positive) so it has a unique nonnegative square root (Square roots exist: a unique $\sqrt{a}\ge 0$ with $(\sqrt{a}{)}^2=a$; the positives are $\{x^2:x\mathrm{\ne }0\}$); and $\{\mathrm{\mid }{x}_k-y_k\mathrm{\mid }:k<n\}$ is a nonempty finite subset of $\mathbb{R}$, because $n\ge 1$, so it has a maximum (Every nonempty finite set of reals has a maximum and a minimum, Maximum and minimum of a set).

Then $d_1$, $d_2$ and $d_{\mathrm{\infty }}$ are metrics on ${\mathbb{R}}^n$ (Metric space: $d(x,y)=0$ iff $x=y$, symmetry, and the triangle inequality; pseudometric and ultrametric).

Why $n\ge 1$. For $n=0$ the set ${\mathbb{R}}^0$ has exactly one element, the empty function, and $d_1$ and $d_2$ are the empty sum $0$ and its root; but $d_{\mathrm{\infty }}$ would be the maximum of the empty set, which does not exist. The hypothesis $n\ge 1$ is therefore not decoration, and it is carried by every statement about $d_{\mathrm{\infty }}$ in this library.

• ${\mathbb{R}}^n$ is defined ZFC-natively here, as the set of functions from the von Neumann natural $n$ to $\mathbb{R}$, precisely so that its coordinates are indexed by $k<n$ and the finite-sum machinery of Finite sums and finite products, by recursion, Minkowski's inequality for finite sums (rational exponent) and The Cauchy-Schwarz inequality for finite sums, all of which sum over $k<n$, applies without any reindexing.
• No rational power appears anywhere above. The triangle inequality for $d_2$ is obtained from Cauchy-Schwarz and the existence of square roots, not from Minkowski at $p=2$, so this lemma does not depend on the theory of rational exponents. Minkowski is used only at $p=1$, where its statement is the termwise sum of the two-term triangle inequality.
• The three metrics are Lipschitz equivalent, with explicit constants, and in particular have the same topology; that computation is on the companion page and is not needed here.