I need help with the following problem in analysis:
Find a sequence of closed intervals I_1 I_2 ... I_n ... whose end points are rational numbers and such that I_n = {e}.
I feel certain that {e} is a set with one element, though the problem doesn't specifically state that is the case. However, our text does use the normal symbol as the empty set, so I imagine with the context of this section added, we are talking about a set with one element. Thank you so much!
Now I am really confused... I think that e (the exponential function) is supposed to be contained in each of the I_n's, and it is the only element that is contained in all of the I_n's. But I am not sure how to define a set of closed intervals such that the intersection of those intervals is e. Thank you.
Of course NOT.
For one, is not a closed interval, much less with rational endpoints.
The whole point of this question is this: e is not a rational number.
BUT e is the limit of an increasing sequence of rational numbers and the limit of an decreasing sequence of rational numbers.