For $a,b,c,d \in [-1,1]$, is $|ab-cd| \le |a-c| + |b-d|$?

Question

The title pretty much says it. Suppose $a,b,c,d \in [-1,1]$. I wish to prove the inequality $$|ab-cd| \le |a-c| + |b-d|.$$ This must be very elementary, but for some reason I'm stuck. Various attempts to rearrange or square both sides didn't seem helpful.

This inequality arises in showing a "Leibniz rule" property for Dirichlet forms.

Thanks!

score 3 · Accepted Answer · answered Apr 25 '13 at 01:24

3

Add and subtract a cross term:

$$|ab-cd|=|ab-ad+ad-cd| \leq |a||b-d|+|d||a-c| \,.$$

answered Apr 25 '13 at 01:24

N. S.

132,525

Bingo! Thanks very much, you saved me a couple hours of staring. – Nate Eldredge Apr 25 '13 at 01:28

For $a,b,c,d \in [-1,1]$, is $|ab-cd| \le |a-c| + |b-d|$?

1 Answers1