Upper bound = maximum

Since $a, b, c$ are in the interval $[0,1]$ then the supremum or the least upper bound is a number $s$ where every element in the interval $[0,1]$ should be less than $s$ . Thus, the upper bound is $2$

If we consider $S$ as the perimeter of a triangle, then $S= a+b+c$ .

We know that the sum of two sides is greater than half the perimeter which is $\frac{S}{2}\implies$

$2(a+b)>S$

$2(c+a)>S$

$2(c+b)>S$

$\implies$ $\large\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b} +\frac{2(a+b+c)}{(a+b+c)}=2$