Chapter 2 Sec. 2.6 Hoffman Kunze Linear Algebra exercise 1

Chapter 2 Sec. 2.6 Hoffman Kunze Linear Algebra exercise 1

Let $s<n$ and $A$ an $s \times n$ matrix with entries in the field F. Use
theorem 4(not its proof) to show that there is a non-zero $X$ in $F^{n
\times 1}$ such that $AX=0$.
Theorem 4
Let V be a vector space which is spanned by a finite set of vectors
$a_{1},a_{2},...,a_{n}$. Then any independent set of vectors in V is
finite and contains no more than n elements.
The problem is straightforward, I think, $AX=0$ denotes a system that is
equivalent to a system denoted by $BX=0$, where $B$ is exactly as $A$
except that it has $s-n$ zero rows at the bottom. $B$'s RREF has at least
one zero row at the bottom and as a consequence at least one variable will
be free.
How to solve this using theorem 4?