I'm having a hard time understanding this proof.
for each point of .
each open interval of the form , where
Contains ration and irrational .
Ok so far, but then
Considering and , we have and , by the squeeze rule.
I don't understand what's going on here. As there are only two sequences, how are these two sequences being squeezed?
I'm sorry, I'm still not sure I understand. We have a second sequence defined on a narrower interval. We are then saying this sequence is squeezed by the interval on an interval of which the interval of is a subset. But this only saying that if then does, how does this tell us ?
Also, why less and not as the sequence can never be more than that value from ?
I'm not sure I see. But here 's my understanding so far. Each rational/irrational number in each interval we define gets closer to the point on which we have defined the interval the narrower we make the interval. So a sequence on the larger interval will squeeze the sequence on the smaller interval. But I'm not sure how this helps. Are we saying something like we are creating a sequence of intervals and then extracting numbers from within those intervals to create a sequence of numbers that tend to the point around which the interval is defined?