Lebesgue Measure on Euclidean Space is Sigma-Finite

In the setting of Euclidean Space and Lebesgue Measure: Standing Notation, let $q\in\mathbb{N}$ with $1\le q$, let $o\in\mathbb{R}^{q}$ be the zero vector of the real vector space $\mathbb{R}^{q}$, and let $\iota:\mathbb{N}\to\mathbb{R}$ be the canonical map; for $m\in\mathbb{N}$ the open ball $B(o,\iota(m))$ is defined, since $0<\iota(m)$ by claim 3 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field.

($\sigma$-finiteness)¶ For every $m\in\mathbb{N}$ the ball $B(o,\iota(m))$ belongs to $\mathcal{B}(\mathbb{R}^{q})$ and satisfies $\lambda_{q}(B(o,\iota(m)))<\infty$, and $\mathbb{R}^{q}=\bigcup_{m\in\mathbb{N}}B(o,\iota(m))$. Hence $\lambda_{q}$ is $\sigma$-finite.