Problem
Prove that $V$ is infinite-dimensional if and only if there is a sequence $v_1, v_2, \dots$ of vectors in $V$ such that $v_1, \dots, v_m$ is linearly independent for every positive integer $m$
Solution
$(\rightarrow)$ $V$ being infinite-dimensional implies that we can always find a $v_{m+1}$ that is not in the span of linearly independent vectors $v_1, \dots, v_m$ for all positive integers $m$. By P2A11 we can therefore construct a sequence $v_1, v_2, \dots$ of vectors with the property that for any $m \in \mathbb{N}$ we have linear independence for $v_1, v_2, \dots, v_m$.
$(\leftarrow)$ Given that we have an infinite sequence of linearly independent vectors in $V$ theorem 2.23 implies that no finite list of vectors will span the space, therefore $V$ is infinite dimensional.