<s>[INST] Prove : $\frac{1}{a}+\frac{1}{b}-\frac{1}{c}< \frac{1}{abc}$ $a;b;c>0$ such that $a^2+b^2+c^2=\frac{5}{3}$. Prove :
$\frac{1}{a}+\frac{1}{b}-\frac{1}{c}< \frac{1}{abc}$
 [/INST] Hint: $(a+b-c)^2 \ge 0 \implies (a^2+b^2+c^2) \ge 2(bc + ca - ab)$
Can you relate both sides to the question at hand?
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