They must continue playing. There is no termination condition for the game. They always continue playing an infinite number of steps.
Alice picks s2. Then Bob picks some number less than s3, so the sequence does not converge to s3. Then Alice picks some number larger than s2, so the sequence does not converge to s2. Any number that is picked after a finite number of steps, cannot be the value that the sequence converges to.
Ok, that makes more sense. The part I still don't get is how we can pick a number x such that s2 < x < s3 without directly assuming that [0, 1] is uncountable.
Even the rational numbers have that property (the Archimedean property): for any distinct rationals s2 and s3, there exists a rational x such that s2 < x < s3.
Proof: x = (s2 + s3) / 2.
This is easily extended to show that there is an arbitrary number of such x.
Alice picks s2. Then Bob picks some number less than s3, so the sequence does not converge to s3. Then Alice picks some number larger than s2, so the sequence does not converge to s2. Any number that is picked after a finite number of steps, cannot be the value that the sequence converges to.