1. Abstract Inequalities

Can anyone help me confirm if I've solved this correctly?

Many thanks.

Q.
If $\displaystyle a^2+b^2=1$ & $\displaystyle c^2+d^2=1$, show that ac + bd < 1.

Attempt: $\displaystyle a^2+b^2+c^2+d^2=2$
if $\displaystyle a^2+c^2\geq2ac$ & $\displaystyle b^2+d^2\geq2bd$
if $\displaystyle 2ac+2bd\leq2$
if $\displaystyle ac+bd\leq1$...true

2. Re: Abstract Inequalities

Yes, that is a perfectly good proof.

(Of course, $\displaystyle a^2+ c^2\ge 2ac$ because $\displaystyle (a- c)^2= a^2- 2ac+ c^2\ge 0$.)

3. Re: Abstract Inequalities

Great. Thank you.