Let $f,g: V \rightarrow W$ be linear operators. Prove that $r(f + g) \leq r(f) + f(g)$ Let $f,g: V \rightarrow W$ be linear operators.
Prove that $r(f + g) \leq r(f) + f(g)$ 
Note: r = rank
My idea was to use matrix representation and to prove that the rank of matrix C ($C= F+G$) can't be bigger than the sum of $r(F) + r(G)$,but to use that idea I must prove that $r(f) = r(g)$ which I don't have an idea how to do.
Other idea was to prove that $(f+g)(v) = f(v) + g(v)$ but also with no luck.

Hint: $\operatorname{im}(f+g)\subseteq \operatorname{im}f+\operatorname{im}g$
