Prove that $\frac{bc}{b+c}+\frac{ca}{c+a}+\frac{ab}{a+b}<\frac{a+b+c}{2}$
Since $(b+c)^2 \geq 4bc$, we have $$\dfrac{bc}{b+c} \leq \dfrac{b+c}4$$ Use the same inequality for the rest and obtain what you want.
Prove that $\frac{bc}{b+c}+\frac{ca}{c+a}+\frac{ab}{a+b}<\frac{a+b+c}{2}$
Since $(b+c)^2 \geq 4bc$, we have $$\dfrac{bc}{b+c} \leq \dfrac{b+c}4$$ Use the same inequality for the rest and obtain what you want.