Suppose that $\left\{\mathbf{e}_{1}, \ldots, \mathbf{e}_{r+1}\right\}$ is a linearly independent set of distinct elements of a vector space $V$ and $\left\{\mathbf{e}_{1}, \ldots, \mathbf{e}_{r}, \mathbf{f}_{r+1}, \ldots, \mathbf{f}_{m}\right\}$ spans $V$. Prove that $\mathbf{f}_{r+1}, \ldots, \mathbf{f}_{m}$ may be reordered, as necessary, so that $\left\{\mathbf{e}_{1}, \ldots \mathbf{e}_{r+1}, \mathbf{f}_{r+2}, \ldots, \mathbf{f}_{m}\right\}$ spans $V$.

Suppose that $\left\{\mathbf{e}_{1}, \ldots, \mathbf{e}_{n}\right\}$ is a linearly independent set of distinct elements of $V$ and that $\left\{\mathbf{f}_{1}, \ldots, \mathbf{f}_{m}\right\}$ spans $V$. Show that $n \leqslant m$.