Proof of a comparison inequality I'm working on a problem that's been giving me some difficulty. I will list it below and show what work I've done so far:

If a, b, c, and d are all greater than zero and $\frac{a}{b} < \frac{c}{d}$, prove that $\frac{a}{b} < \frac{a + c}{b + d} < \frac{c}{d}$. 

Alright, so far I think the first step is to notice that $\frac{a}{b+d} + \frac{c}{b+d} < \frac{a}{b} + \frac{c}{d}$ but after that I'm not sure how to continue. Any assistance would be a great help! 

As $$\frac ab<\frac cd\implies ad<bc\text { as } a,b,c,d>0$$
$$\frac{a+c}{b+d}-\frac ab=\frac{b(a+c)-a(b+d)}{b(b+d)}=\frac{bc-ad}{b(b+d)}>0 \text{ as } ad<bc \text{ and }  a,b,c,d>0 $$
So, $$\frac{a+c}{b+d}>\frac ab$$
Similarly,  $$\frac{a+c}{b+d}-\frac cd=\frac{ad-bc}{d(b+d)}<0\implies  \frac{a+c}{b+d}<\frac cd$$
