How do I prove m events are independent if the following is true?

Question

I can prove it for $2$ events however for $m$ arbitrary events I can't so I was wondering if there is a proof of some sort for the following?

Independence of $m$ events: Similarly, $m$ events $A_1, A_2, \cdots, A_m$ are called independent if $$\mathbb P\left(A_1\cap A_2\cap\cdots\cap A_m\right) = \mathbb P\left(A_1\right)\cdot\mathbb P\left(A_2\right)\cdot\cdots\cdot\mathbb P\left(A_m\right)$$

Answer 1

The formulation “are called” indicates a definition. That’s just the definition of independence of events, so there’s nothing to prove.

Answer 2

Make an inductive argument, taking a new event $$\mathbb P(B) = \mathbb P(A_{i} \cap A_{i+1})$$ then showing that intersects with your k-th element using the 2 event proof.