Suppose a firm produces Q units using labour (L) for which the wage, w>0 and the price of capital (K) is r>0. It cannot employ negative amounts of its factors: L $\geq 0$, K $\geq 0$.
The firm's constraint can be written as $wL + rK$ $\leq Q$
How does one prove that such a set is convex? We know that if a constraint is convex, some vector, z = $\lambda x + (1-\lambda)y$ lies within the set.
Suppose I choose $$z_1 = \lambda L_1 + (1-\lambda) L_2 $$
$$z_2 = \lambda K_1 + (1-\lambda) K_2 $$
where $\lambda \exists [0,1]$
z = L + K
$$wz_1 + rz_2 =w[\lambda L_1 + (1-\lambda) L_2] + r[\lambda K_1 + (1-\lambda) K_2] $$ $$=\lambda[wL_1 + rK_1] +(1-\lambda)[wL_2 + rK_2]$$ $$\leq \lambda Q + (1-\lambda)Q$$ $$= Q$$