<s>[INST] Prove that $\frac{a}{b} < \frac{a+c}{b+d} <\frac{ c}{d}$ I'm reading a introductory book on mathematical proofs and I am stuck on a question.

Let $a, b, c, d$ be positive real numbers, prove that if $\frac{a}{b} < \frac{c}{d}$, then $\frac{a}{b} < \frac{a+c}{b+d} <\frac{ c}{d}$.

 [/INST] $$\frac{a}{b}=\frac{a(1+\frac{d}{b})}{b(1+\frac{d}{b})}=\frac{a+d(\frac{a}{b})}{b+d}<\frac{a+d(\frac{c}{d})}{b+d}=\frac{a+c}{b+d}$$
and similarly for the other inequality.
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