If an infinite sequence of all heads is a possible outcome, then it is not true that the relative frequency converges to 0.5 in the limit. It is obviously true for many infinite sequences but if all heads is a possibility then it is not true in general.

Just to clarify, I am talking about the frequentist's claim that in the long run the relative frequency becomes the probability, i.e. P(x) = lim[n -> ∞] nx / n where n is the number of trails and nx is the number of trails that yielded x. So again, if an infinite sequence of heads is a possible outcome, then that would mean P(heads) = 1 while the frequentist asserts that it necessarily must become 0.5.

Does the frequentist view really claim that infinite sequence nx/n always converges, for all possible runs? That is obviously false. The laws of large numbers only state that probability that the limit of nx/n is equal to 1/2 is 1. This is because all sequences are equally probable and there are overwhelmingly many more sequences that limit to 1/2 than sequences that do not. That is different from saying that the limit is always equal to 1/2, because there are these exceptional cases of low measure like HHHHH..., HTTHTTHTT...