**Lemma** Take $a,b,c,d \in \mathbb{R}$. If $0<a<b$ and $0<c<d$ then $ac<bd$ See application [[Inequalities for powers of positive reals with natural number exponents]]. **Proof** Since $c>0$ multiplying the first identity by $c$ gives $ac<bc$ using [[Real numbers#^2d1ba9|axiom]]. Similarly multiplying the second by $b$ gives $bc<bd$. Hence, using [[Real numbers#^2d1ba9|transitivity]], $ac<bd$.