Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$

Let $n$ be a natural number, let $x=(x_1,\dots,x_n)$, $y$ and $h$ be points of Euclidean space $\mathbb{R}^n$, and let $\lambda$ be a real number. The real numbers form an ordered field, with additive identity $0$ and order $\le$; for a real number $t$ write $t^2$ for $t\cdot t$ and $|t|$ for its absolute value. Sums $\sum_{i=1}^{n}$ below are finite sums in the field of real numbers, and index ranges such as $1\le i\le n$ use the order on the natural numbers.

Write $\lVert\,\cdot\,\rVert$ for the Euclidean norm, $d_E$ for the Euclidean distance, $0_{\mathbb{R}^n}$ for the origin of $\mathbb{R}^n$, $x+y$ for the sum of points, $\lambda x$ for the scalar multiple, and $x-y$ and $x\cdot y$ for the difference and dot product of points of $\mathbb{R}^n$. Then the following hold.

1. (Square of the norm)¶ $\lVert x\rVert$ is the unique real number $r$ with $0\le r$ and $r^{2}=\sum_{i=1}^{n}x_i^{2}$. In particular

$$0\le\lVert x\rVert,\qquad \lVert x\rVert^{2}=\sum_{i=1}^{n}x_i^{2}=x\cdot x .$$

2. (Distance)¶ $d_E(x,y)=\lVert x-y\rVert$; in particular $d_E(x,0_{\mathbb{R}^n})=\lVert x\rVert$ and $d_E(x,x+h)=\lVert h\rVert$.

3. (Vanishing)¶ $\lVert x\rVert=0$ if and only if $x=0_{\mathbb{R}^n}$.

4. (Coordinate bound)¶ $|x_i|\le\lVert x\rVert$ for every natural number $i$ with $1\le i\le n$.

5. (Absolute homogeneity)¶ $\lVert\lambda x\rVert=|\lambda|\,\lVert x\rVert$.

6. (Triangle inequality)¶ $\lVert x+y\rVert\le\lVert x\rVert+\lVert y\rVert$.