How can I prove that for a production function $F:\mathbb X \rightarrow \mathbb R$ with constant returns to scale
$$\forall x\in \mathbb X, \forall t > 0: \ \ F(tx) = t F(x)$$
and with the cost function
$$C(p,y):=\min_{x \in \mathbb X} \{p^\top x\lvert F(x) \geq y\}$$
it must be the case that $$C(p,ty) = tC(p,y).$$
My first strategy has been to make a constructive proof along the following lines
$$C(p,ty) = \min_{x \in \mathbb X} \{p^\top x\lvert F(x) \geq ty\} =\min_{x \in \mathbb X} \{p^\top x\lvert F(x/t) \geq y\},$$ with the second identity using the constant returns to scale assumption. I then define $z=x/t$ and rewrite the variable over which to minimize $$= \min_{x/t} \{t p^\top(x/t) \lvert F(x/t) \geq y\} = t \min_z\{p^\top z \lvert F(z) \geq y\} = t C(p,y).$$
However I am not entirely sure that this proof is solid so I guess my question boils down to whether this proof really is solid or if there is another way to proove it?
