I am trying to prove this question, but I am not sure if what I did is correct..

q: Suppose that A1, A2, ..., An are events in a random experiment whose intersection has positive probability. Prove the multiplication rule of probability.

This is what I have done:

[by the definition of conditional prob]

now assuming this is true for all n, we have

thus it is true for all n..

Is it path I have followed correct? Any inputs will be appreciated!

Thanks