Let VV be a vector space and F\mathbf F denote R\mathbf{R} or C\mathbf{C} and let f:V→Ff : V \to \mathbf F be defined such that f(v+w)=f(v)+f(w)f(v+w) = f(v)+f(w).

Consider a ratio of naturals n/mn/m where n,m∈Nn,m \in \mathbf{N} - we have

m⋅f((n/m)v)=f((n/m)v)+⋯+f((n/m)v)⏟m times=f(nv)m \cdot f((n/m)v) = \underbrace{f((n/m)v) + \dots + f((n/m)v)}_{\text{$m$ times}} = f(nv)

Implying f((n/m)v)=(n/m)f(v)f((n/m)v) = (n/m)f(v). Negatives are easy to see by additivity

f(−(n/m)v)+f((n/m)v)  ⟹  f(−(n/m)v)=−(n/m)f(v)f(-(n/m)v) + f((n/m)v) \implies f(-(n/m)v) = -(n/m)f(v)

Thus f(rv)=rf(v)f(rv) = rf(v) for all r∈Qr\in \mathbf{Q}.