Let’s look at the ultimate extreme version. Assume she’s woken up once (or arbitrarily many non-zero times) for tails, and not at all for heads. Now the fact that she’s been woken up implies tails with certainty. So if the answer remains 1⁄2 in the extreme versions, then there must be a discontinuous jump, rather than convergence, when the ratio of the number of awakenings for heads vs. tails tends towards zero.
Let’s look at the ultimate extreme version. Assume she’s woken up once (or arbitrarily many non-zero times) for tails, and not at all for heads. Now the fact that she’s been woken up implies tails with certainty. So if the answer remains 1⁄2 in the extreme versions, then there must be a discontinuous jump, rather than convergence, when the ratio of the number of awakenings for heads vs. tails tends towards zero.
Intuitively, this is very convincing. But it definitely doesn’t prove anything by itself...