Why is $c^3\ge abc$?

Question

Why is $c^3\ge abc$ ?

there is a step in proving some inequality which I don't understand. It is clear that $a^3+b^3\ge ab^2+ba^2=ab(a+b)\tag1$ from rearrangement inequality, but then how does he conclude that the whole $a^3+b^3+c^3\ge ab(a+b)+abc$. He should have proved (assumed) that somehow directly, because otherwise he wouldn't show $(1)$

Answer 1

I think it's a typo.

It should be $\frac{abc}{a^3+b^3+abc}\leq\frac{abc}{ab(a+b)+abc}=\frac{c}{a+b+c}$