Convergence in Euclidean Space is Coordinatewise Convergence

Let $n$ be a natural number, let $[n]$ be the initial segment of $\mathbb{N}$ determined by $n$, and let $\mathbb{R}^n$ be Euclidean space equipped with the Euclidean distance $d_E$, which is a metric by Euclidean Distance is a Metric on $\mathbb{R}^n$. Let $(\mathbb{R},d_{\mathbb{R}})$ be the real line.

Let $(x^{(m)})_{m\in\mathbb{N}}$ be a sequence in $\mathbb{R}^n$ and let $x\in\mathbb{R}^n$, with coordinates written $x^{(m)}=(x^{(m)}_1,\dots,x^{(m)}_n)$ and $x=(x_1,\dots,x_n)$.

Then $(x^{(m)})_{m\in\mathbb{N}}$ converges to $x$ in $(\mathbb{R}^n,d_E)$ if and only if, for every $i\in[n]$, the sequence $(x^{(m)}_i)_{m\in\mathbb{N}}$ converges to $x_i$ in $(\mathbb{R},d_{\mathbb{R}})$.