Working on the last inequality, we have
So we need a lower bound for . Remembering that we are letting x get very close to -1, so that the distance between them is small, we can choose a value that will make that distance small (and we're going to make it smaller anyway). It doesn't matter what value we choose, as long as it is sufficiently close. So say we let . That means
Therefore we can say .
So an appropriate is .