Show $\frac{3997}{4001}>\frac{4996}{5001}$ I wish to show that $$\frac{3997}{4001}>\frac{4996}{5001}.$$
Of course, with a calculator, this is incredibly simple. But is there anyway of showing this through pure analysis? So far, I just rewrote the fractions:
$$\frac{4000-3}{4000+1}>\frac{5000-4}{5000+1}.$$

Using long multiplication we get
$$3997\times5001=19988997>19988996=4996\times4001$$
which implies the desired result (because positive multiplication preserves order).
