Prove linear independence of roots of square-free numbers over $\mathbb{Q}$ using induction

Question

I want to prove that this system $\{1,\sqrt{m_1},\sqrt{m_2},...,\sqrt{m_n}\}$, where all $m_i$ are different square-free natural numbers, is linear independent over $\mathbb{Q}$. My teacher has asked me to prove this using induction.

1. BASE CASE: $a_11+a_2\sqrt{m_1}$ linear independent. It's obviously.
2. STEP CASE: Supouse $\sum\limits_i^{n-1}a_i\sqrt{m_i}\neq0$ for any $a_i\in\mathbb{Q}$.

Now I need to show that $\sum\limits_i^{n}a_i\sqrt{m_i}\neq0$. Everything I can do next is to rewrite it like this(since $a_n\neq0$): $\sqrt{m_n}=\sum\limits_i^{n-1}b_i\sqrt{m_i}$.

But I can't make any contradiction.

EDIT: $m_i\gt1, \forall i$