Problem: Proof that cubic function is injective

Problem

In this problem we will prove formally that

$$a^3=b^3⟺a=b$$

The “ $⟸$ ” part is trivial so we focus on $a^3=b^3⟹a=b$ . To prove this, we begin by proving

$$a>b⟹a^3>b^3$$

or

$$a>b⟹a^3-b^3>0$$

Show that

$$a^3-b^3=\left( a-b \right)\left( a^2+ab+b^2 \right)$$

Show that

$$a^2+ab+b^2= \frac{1}{2}\left( a^2+b^2+{\left( a+b \right)}^2 \right)$$

Show that

$$a>b⇒a^3-b^3>0$$

Prove that

$$a^3=b^3⟺a=b$$