I was asking myself why is a Giffen good an inferior good and I read the following post : Difference between Giffen and inferior goods. Why aren't all inferior goods Giffen goods?.
The 2nd comment of @Matthew Gunn presents things very well. A good is called inferior if you purchase less as your income increases: $\frac{dq(p,w)}{dw}<0$ where $p$ is its price and $w$ the income of the consumer.
A good is called a Giffen good if you purchase more as its price increases : $\frac{dq(p,w)}{dp}>0$.
I cannot intuitively and mathematically see why a Giffen good must be an inferior good : $\frac{dq(p,w)}{dp}>0$ does not imply that $\frac{dq(p,w)}{dw}<0$. Also, if we take an example of a very poor community that cannot afford meat so that potatoes become a Giffen good. Then if the incomes of a consumer increase, nothing guarantees that he will purchase less potatoes, right ? Maybe the increase in his income is so small that he will not be able to afford meat and keep buying more potatoes, so they are Giffen goods but not inferior.
Can someone prove that $\{\text{Giffen goods} \} \subsetneq \{\text{Inferior goods}\}$ ?
(I am new to microeconomics sorry if I am dealing with this the wrong way).