For any positive numbers $x_1$, $x_2$, $y_1$, $y_2$, if $\frac{x_1}{y_1} < \frac{x_2}{y_2}$ , then $\frac{x_1}{y_1} < \frac{x_1+x_2}{y_1+y_2} < \frac{x_2}{y_2}$

    在國際貿易理論中，你會看著世界(國內加國外)的商品相對產量，會介於國內kap國外相對產量之間。以數學表示，若$\frac{x_1}{y_1} < \frac{x_2}{y_2}$，煞$\frac{x_1}{y_1} < \frac{x_1+x_2}{y_1+y_2} < \frac{x_2}{y_2}$。
In international trade theory, you will see that the world's (home plus foreign) relative goods quantity is between homes and foreigns. Mathematically speaking, if $\frac{x_1}{y_1} < \frac{x_2}{y_2}$, then $\frac{x_1}{y_1} < \frac{x_1+x_2}{y_1+y_2} < \frac{x_2}{y_2}$.
#PaulKrugman #MauriceObstfeld #MarcMelitz

$ \frac{x_1}{y_1} < \frac{x_2}{y_2} \Rightarrow \frac{x_1+x_2}{y_1+y_2} < \frac{x_2}{y_2} :$
$$\frac{x_1}{y_1} < \frac{x_2}{y_2}$$
$$ \Rightarrow \frac{x_1+x_2}{y_1} = \frac{x_1}{y_1} + \frac{x_2}{y_1} < \frac{x_2}{y_2}+ \frac{x_2}{y_1}$$
$$ \Rightarrow \frac{x_1+x_2}{y_1} < \frac{x_2 y_1 + x_2 y_2}{y_2 y_1}$$
$$ \Rightarrow x_1+x_2 < \frac{x_2 (y_1 + y_2)}{y_2}$$
$$ \Rightarrow \frac{x_1+x_2}{y_1+y_2} < \frac{x_2}{y_2} $$

$ \frac{x_1}{y_1} < \frac{x_2}{y_2} \Rightarrow \frac{x_1}{y_1} < \frac{x_1+x_2}{y_1+y_2} :$
$$\frac{x_1}{y_1} < \frac{x_2}{y_2}$$
$$ \Rightarrow \frac{x_1}{y_1} + \frac{x_1}{y_2} < \frac{x_2}{y_2} + \frac{x_1}{y_2} = \frac{x_1+x_2}{y_2}$$
$$ \Rightarrow \frac{x_1 y_2 + x_1 y_1}{y_1 y_2} < \frac{x_1+x_2}{y_2}$$
$$ \Rightarrow \frac{x_1 (y_1 + y_2)}{y_1} < x_1+x_2$$
$$ \Rightarrow \frac{x_1}{y_1} < \frac{x_1+x_2}{y_1+y_2} $$

kā頂面兩个證出來的結果合做伙，就故得證囉。
Combining the results of the above two proofs, QED.