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

\[a^3=b^3⟺a=b\]