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

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

Q.

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