\[\mathbf{A}\]
with differentiable components defined on a volume \[V\]
with boundary \[S\]
\[\int \int \int_V \mathbf{\nabla} \cdot \mathbf{A} dV = \int \int_S \mathbf{A} \cdot \mathbf{n} dS \]
Let
\[\mathbf{A} = \mathbf{\nabla} \times \mathbf{F}\]
for some vector field \[\mathbf{F}\]
Then
\[\int \int \int_V \mathbf{\nabla} \cdot (\mathbf{\nabla} \times \mathbf{F}) dV = \int \int_S (\mathbf{\nabla} \times \mathbf{F}) \cdot \mathbf{n} dS \]
Then the right hand side is zero. Hence
\[\int \int \int_V \mathbf{\nabla} \cdot (\mathbf{\nabla} \times \mathbf{F}) dV =0 \]
The surface is arbitrary and so is the volume, hence
\[\mathbf{\nabla} \cdot (\mathbf{\nabla} \times \mathbf{F}) =0\]