There is no circularity. You're discussing the real-world interpretation of the probability of an event in terms of long-term frequencies. The probability in the definition of convergence is a well-defined mathematical concept that has nothing to do with frequencies.

Edit: thinking more about it, I agree that the passage from mathematical probability to real-world probability is not very satisfactory. But it's not really surprising, the very notion of "real-world" is troublesome.