Let $a \in \mathbb{Q}_3$ and suppose that $\vert a-1 \vert_3 \leq 3^{-2}$. Show that $a \in {\mathbb{Q}_3}^3$ using Hensel's Lemma.
My idea is the following: I consider $f(x) = x^3 - a$ and want to apply Hensel to deduce the claim. Observe that: $f'(x) = 3x^2$, $\vert f(1)\vert_3 \leq 3^{-2}$ and $\vert f'(1) \vert_3 = 3^{-1}$. However, the condition in Hensel's Lemma ($\vert f(1)\vert_3 < {\vert f'(1)\vert_3}^2$) does not hold. Could someone give me a hint on how to apply Hensel in this setup?