Do perfect complements have to be normal goods? If so, why? Two goods $x,y$ are perfect complements if they have the utility function 
$$U(x,y) = \min \lbrace ax,by \rbrace $$ 
$$a,b \in \Bbb{Q}^+$$
My professor said $x,y$ have to be normal goods but didn't explain why well enough that I could understand. 

My Question: 

Are perfect complements always normal goods? If so, why?
A good is normal if its demand is increasing in income. So let $p_x$ and $p_y$ be the price of the goods with quantities $x$ and $y$ and let $m$ be income.

Suppose $ax&gt;by$. Then $\min\{ax,by\}=by$. By slightly reducing $x$ by and spending the saved money on $y$, one gets a better bundle. For an optimal bundle, this cannot be.  

Similarly, it cannot be optimal that $by&gt;ax$. So in the optimal consumption bundle, it must be the case that $ax=by$. It is also not that hard to see that the consumer will spend all her income. So rewrite the condition as 
$$y=\frac{a}{b}x$$ and plug it into the budget equation $$p_x x+p_y y=m$$ to get
$$p_x x+ p_y\frac{a}{b}x=m=x\Big(p_x+p_y\frac{a}{b}\Big).$$
Therefore, we get the demand function given by $$x(p_x,p_y,m)=\frac{m}{p_x+p_y\frac{a}{b}},$$ which is clearly increasing in $m$. Similarly, one shows that the other good is normal too.

Pedantic remark: A differentiable function can be increasing at every point without the derivative being strictly positve everywhere. The function given by $x\mapsto x^3$ has derivative $0$ at $0$ but is everywhere increasing.