Sorry if this is a trivial question.
The book is Linear Algebra Done Right by Axler, page 25-26.
Theorem: In a finite-dimensional vector space, the length of every linearly independent list of vectors is less than or equal to the length of every spanning list of vectors.
Proof: Suppose that $(u_1 ,\ldots, u_m)$ is linearly independent in $V$ and that $(w_1,\ldots ,w_n)$ spans V. We need to prove that $m \leq n$. We do so through the multistep process described below; note that in each step we add one of the $u$'s and remove one of the $w$'s.
Step 1: The list $(w_1,\ldots, w_n)$ spans $V$, and thus adjoining any vector to it produces a linearly dependent list. In particular, the list $(u_1,w_1, \ldots,w_n)$ is linearly dependent.
Question: Why is $(u_1,w_1, \ldots,w_n)$ is linearly dependent?