Let $V$ be a vector space over the field $\mathbb{F}$ and let's define the properties $(f)$ and $(i)$ in the following way:
$f$) $\exists F \in V$ finite such that $\text{span}(A) = V$
$i$) $\exists I \in V$ infinite such that $I$ is lineary independent
If I now define a vector space to be finite-dimensional if it satisifes $(f)$ and infinite dimensional if it satifies $(i)$, I definitely need to make sure that every vector space is either finite-dimensional or infinite-dimensional and cannot be both.
So the question is how to prove that
$$
f \iff \overline{i}
$$
or equivalently that
$$
i \iff \overline{f}
$$
Best Answer
Hint: if (f) fails, you can inductively construct a linearly independent sequence $x_1, x_2, \dots$.
If $x_1, \dots, x_{n-1}$ have been chosen, then by assumption their span is not equal to $V$, so you can choose an $x_n$ which is not in their span...
Conversely, suppose $f$ holds, so that there is a finite set, call it $F = \{x_1, \dots, x_n\}$ which spans $V$. If $I$ is an infinite set, you can find $n+1$ distinct elements $y_1, \dots, y_{n+1}$ in it. By elementary linear algebra, they cannot be linearly independent.