Vector model of atom. Hopefully easy question.

[LagrangeEuler]

In system with one electron total angular moment vector $\vec{j}$ is just:
$$\vec{j}=\vec{l}+\vec{s}$$
http://selfstudy.in/MscPhysics/BScVectorModelOfAtom.pdf
In page 3
author draw a triangle. Intensities of the vectors are $|\vec{l}|=\sqrt{l(l+1)}\hbar$, $|\vec{s}|=\sqrt{s(s+1)}\hbar$, $|\vec{j}|=\sqrt{j(j+1)}\hbar$
And then from nowhere $j=l+s$ or $j=l-s$. Could you please explain me that! Tnx.

 

[DrDu]

These are maximal and minimal possible values, respectively. If you add two vectors, the length of the sum will always lie between these two extremal values.

 

[LagrangeEuler]

But if I look this intensity formulas
$$|\vec{l}|+|\vec{s}|\neq |\vec{j}|_{for j=l+s}$$
Right?