Tangent identifies a bounded incomplete interval with the unbounded complete real line

Given: The interval $I:=(-\pi \mathrm{/}2,\pi \mathrm{/}2)$ and the usual absolute-value metrics on $I$ and $\mathbb{R}$.

Tangent restricts to a continuous strictly increasing bijection $\tan :I\to \mathbb{R}$, whose inverse $\arctan :\mathbb{R}\to I$ is continuous (The principal inverse tangent $\arctan :\mathbb{R}\to (-\pi \mathrm{/}2,\pi \mathrm{/}2)$).

A homeomorphism is a continuous bijection with continuous inverse (Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological).

In the usual metric, $B(0,r)=(-r,r)$ for $r>0$ (The absolute value makes $\mathbb{R}$ a metric space: $d(x,y)=\mathrm{\mid }x-y\mathrm{\mid }$ is a metric, its open balls are the intervals $(x-r,x+r)$, and it is unbounded).

Every Cauchy sequence of real numbers converges to a real number (The reals are complete).

For a subset of $\mathbb{R}$ with the subspace metric, metric Cauchy sequences are exactly real Cauchy sequences (Dictionary: for $A\subseteq \mathbb{R}$ with the metric $d(x,y)=\mathrm{\mid }x-y\mathrm{\mid }$, continuity and uniform continuity of $f:A\to \mathbb{R}$ agree with the metric-space notions, the Lipschitz and Hölder conditions are the metric ones instantiated, and a subset of $\mathbb{R}$ is compact in the open-cover sense of $\mathbb{R}$ exactly when it is a compact metric subspace, claim 7).

A subset of a metric space is closed exactly when it is sequentially closed (A point lies in the closure of $A$ iff some sequence in $A$ converges to it, and a set is closed iff it is sequentially closed, claim 2).

A subspace of a complete metric space is complete if and only if it is closed (A subspace of a complete metric space is complete iff it is closed, and a complete subspace of any metric space is closed).

For every real $\epsilon >0$, there is a positive integer $N$ with $1\mathrm{/}N<\epsilon$ (For every $\epsilon >0$ in a complete ordered field there is a natural $n\ge 1$ with $1\mathrm{/}n<\epsilon$).

The number $\pi =2\gamma$ is positive because the smallest positive zero of cosine satisfies $\gamma \in (0,2)$ (Pi as twice the smallest positive zero of cosine, Cosine has a smallest positive zero, lying strictly between zero and two).

The usual real line is not a bounded metric space (The absolute value makes $\mathbb{R}$ a metric space: $d(x,y)=\mathrm{\mid }x-y\mathrm{\mid }$ is a metric, its open balls are the intervals $(x-r,x+r)$, and it is unbounded).

A sequence converges in a metric space when its distance from the proposed limit is eventually below every positive tolerance (Convergence of a sequence in a metric space: $x_k\to x$ iff $d(x_k,x)\to 0$ in $\mathbb{R}$); for the usual real metric this distance is $\mathrm{\mid }{x}_k-x\mathrm{\mid }$ (The absolute value makes $\mathbb{R}$ a metric space: $d(x,y)=\mathrm{\mid }x-y\mathrm{\mid }$ is a metric, its open balls are the intervals $(x-r,x+r)$, and it is unbounded).

A metric space is complete when every Cauchy sequence in it converges to a point of the space (Complete metric space: every Cauchy sequence converges in the space).

A subset of a metric space is bounded when it is empty or is contained in some open ball (Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space).