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I am trying to prove AM-GM inequality in $3$ variables.

Prove that $a^3 + b^3 + c^3 \ge 3abc$ for all $a,b,c \in \mathbb{R}^+$.

Could you please verify if my attempt contains logical gaps/errors?


My attempt:

Lemma: $a^2 + b^2 \ge 2ab$ for all $a,b \in \mathbb{R}^+$.

It follows from our lemma that $a^3 + b^3 \ge 2 \sqrt{a^3b^3} = 2ab\sqrt{ab}$ and $c^3 +abc \ge 2\sqrt{c^4ab} = 2c^2 \sqrt{ab}$, and that $ab\sqrt{ab} + c^2 \sqrt{ab} \ge 2\sqrt{ab\sqrt{ab}c^2 \sqrt{ab}} = 2abc$.

As a result, $(a^3 + b^3) + (c^3 +abc) \ge 2ab\sqrt{ab} + 2c^2 \sqrt{ab} \ge 4abc$ and so $a^3 +b^3 +c^3 \ge 3abc$. This completes the proof.

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Hint: Better is to use that $$a^3+b^3+c^3-3abc= \left( a+b+c \right) \left( {a}^{2}-ab-ac+{b}^{2}-bc+{c}^{2} \right) $$

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