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

In the setting of Metric Spaces: Balls, the Metric Topology, and the Real and Complex Numbers as Metric Spaces, let $n\in\mathbb{N}$, with the operations of Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations (iv), the dot product, the Euclidean norm $\lVert\cdot\rVert$ and the Euclidean distance $d_{E}$ on $\mathbb{R}^{n}$. For a sequence $(x^{(m)})$ in $\mathbb{R}^{n}$ and $k\in[n]$, $x^{(m)}_{k}$ is the $k$-th component of $x^{(m)}$, as in Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations (ii); $(\mathbb{R}^{n},d_{E})$ is a metric space by the clause metric below.

(i) $d_{E}$ is a metric on $\mathbb{R}^{n}$.

(ii) For a sequence $(x^{(m)})$ in $\mathbb{R}^{n}$ and $x\in\mathbb{R}^{n}$: $(x^{(m)})$ converges to $x$ in $(\mathbb{R}^{n},d_{E})$ if and only if $(x^{(m)}_{k})$ converges to $x_{k}$ in $(\mathbb{R},d_{\mathbb{R}})$ for every $k\in[n]$.

(iii) A sequence $(x^{(m)})$ in $\mathbb{R}^{n}$ is Cauchy in $(\mathbb{R}^{n},d_{E})$ if and only if $(x^{(m)}_{k})$ is Cauchy in $(\mathbb{R},d_{\mathbb{R}})$ for every $k\in[n]$.

(iv) $(\mathbb{R}^{n},d_{E})$ is complete.

(v) For every $k\in[n]$ the map $x\mapsto x_{k}$, and the map $x\mapsto\lVert x\rVert$, from $(\mathbb{R}^{n},d_{E})$ to $(\mathbb{R},d_{\mathbb{R}})$ are Lipschitz with constant $1$.

(vi) If $(x^{(m)})$ and $(y^{(m)})$ are sequences in $\mathbb{R}^{n}$ converging to $x,y\in\mathbb{R}^{n}$ in $(\mathbb{R}^{n},d_{E})$, and $(t_{m})$ is a sequence in $\mathbb{R}$ converging to $t\in\mathbb{R}$ in $(\mathbb{R},d_{\mathbb{R}})$, then the sequences $(x^{(m)}+y^{(m)})$ and $(t_{m}x^{(m)})$ converge to $x+y$ and $tx$ in $(\mathbb{R}^{n},d_{E})$, and $(x^{(m)}\cdot y^{(m)})$ converges to $x\cdot y$ in $(\mathbb{R},d_{\mathbb{R}})$.