Given two numbers $a,b\in (0, \infty)\subset\mathbb{R}$ and two other numbers $c,d\in (0, \infty)\subset\mathbb{R}$ is it sufficient to show that $$ \dfrac{c}{a} < \dfrac{d}{b} $$ in order to prove that $c<d$?
This seems to pass my intuition check of that if you divide by a larger quantity and find that the ratio is larger, the numerator should be larger
My attempt:
$$ a < b \implies \dfrac{a}{b} < 1 $$ Since $$ \dfrac{c}{a} < \dfrac{d}{b} \implies c < \dfrac{ad}{b} < 1 \cdot d \therefore c<d $$
Does this seem correct or is there an edge case I am missing or logical error?
With $a<b$ as a condition, this is correct indeed. If we don't have this condition, $6/3<4/1$ is a counterexample.