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Proof That the Set of Conservative Fields is a Vector Space

Theorem
The set of conservative vector fields defined on a domain  
\[D\]
  is a vector space.
Proof
Let the set of conservative vector fields be represented by  
\[V\]
  then  
\[\mathbf{F}, \: \mathbf{G} \in V\]
  imply there exist functions  
\[f,g\]
  operating on elements of  
\[D\]
  such that  
\[ \mathbf{F} = \mathbf{\nabla} f, \: \mathbf{G} = \mathbf{\nabla} g \]
.
Then  
\[\mathbf{\nabla} (f+g) =\mathbf{F} + \mathbf{G} \in V\]

Let  
\[\alpha \in \mathbb{R}\]
  then  
\[\mathbf{ \nabla} \alpha f = \alpha \mathbf{\nabla} f = \alpha \mathbf{F} \]

The zero field is conservative, taking the value zero everywhere.
Hence the set of conservative vector fields is a vector space.