So $\frac{1}{a}>\frac{1}{b}+\frac{1}{c}$ $bc>ab+ac$ $ac<bc-ab=b(c-a)<b^2$

Blah blah storage, We know that $h_a>h_b+h_c$, where $h_a,h_b,h_c$ are the altitudes of $\Delta$, this is equivalent to $$\frac{2S}{a}>\frac{2S}{b}+\frac{2S}{c}\Longleftrightarrow bc>ab+ac.$$Suppose $ac\geq b^2$, then we would have $c>a+\frac{ac}{b}\geq a+b$, which contradicts the triangle inequality, we are done.

$ac\geq b^2$