<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] HINT: Note that
$$\begin{align*}
\frac{a+c}{b+d}-\frac{a}b&=\frac{a+c}{b+d}\cdot\frac{b}b-\frac{a}b\cdot\frac{b+d}{b+d}\\
&=\frac{b(a+c)-a(b+d)}{b(b+d)}\\
&=\frac{ab+bc-ab-ad}{b(b+d)}\\
&=\frac{bc-ad}{b(b+d)}\;.
\end{align*}$$
You’d like to show that this difference is positive, so you’d like to know that $bc>ad$. Use the fact that $\frac{a}b<\frac{c}d$ to show this.
The other inequality can be handled similarly.
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