prove this inequality relativistic velocity addition law I'm trying to prove that the set of all velocities under the relativistic velocity addition law, form a group. I proved all other properties except this one : let $c$ be a positive real number and $E={]-c,c[}$ prove that : $$\forall(v_1,v_2)\in E^2 \ \ ,\ \ \frac{v_1+v_2}{1+\dfrac{v_1 \times v_2}{ c^2}} \in E $$

To simplify the notation, we can write $v_i = r_i c.$ Then we want to show that if $r_1,r_2 \in {]-1,1[},$ then $\dfrac{r_1+r_2}{1+r_1r_2} \in {]-1,1[}.$ Now noting that $r_1r_2 \
eq -1,$ we can see that the target inequality is equivalent to $$|r_1 + r_2| < |1 + r_1r_2|$$ Squaring both sides gives $$r_1^2 + 2r_1r_2 + r_2^2 < 1 + 2r_1r_2 + r_1^2r_2^2$$ Moving everything to the right side gives $$0 < 1 - r_1^2 - r_2^2 + r_1^2r_2^2 = (1 - r_1^2)(1 - r_2^2)$$ which is true since $|r_1|,|r_2| < 1.$

Now either just note that every step was reversible (or actually go backwards through the steps to check this) to show what you wanted to show.