A Homogeneous Linear System with More Unknowns than Equations Has a Nonzero Solution

In the setting of Metric Spaces: Balls, the Metric Topology, and the Real and Complex Numbers as Metric Spaces, let $m,n\in\mathbb{N}$ with $m<n$, and let $A\in\mathcal{M}_{m\times n}(\mathbb{R})$ be a real matrix, with the matrix-vector product and the origins $0$ of $\mathbb{R}^{n}$ and $\mathbb{R}^{m}$.

¶ There is $x\in\mathbb{R}^{n}$ with $x\neq0$ and $Ax=0$.