The reason for choosing is that we know with the choice, is still true. That does not mean that there is no other value of of besides . If will work, why do we care about larger value of ? In other words, if the small number is closer to the limit, why bother about other large 's that are farther from limit.
Since we know is true, we know is also true. So we know it is safe to choose , since we know with the choice, will still remain true.
If we know it will work by limiting to to maintain , and we also know between 0 and 1, the is a number that also can maintain being true. Given the choices, of course we would choose over 1, since the smallest we could get is the one that can give us the value closest to the limit.