I'm having trouble grasping this concept. This is the part in question:(adsbygoogle = window.adsbygoogle || []).push({});

Why is the point x uniquely determined by the nested sequence? if i pick two rational numbers, no matter how close together they are, surrounding say the square root of two, shouldn't there always be another two rational numbers that are closer? We require that the length of the interval I_{n}tends to zero with increasing n; that is, that the length of I_{n}is less than any preassigned positive number for all sufficiently large n. A set of closed intervals I_{1}, I_{2}, I_{3}, ... each containing the next one and such that the lengths tend to zero will be called a nested sequence of intervals. The point x is uniquely determined by the nested sequence; that is, no other point y can lie in all I_{n}, since the distance between x and y would exceed the length of I_{n}once n is sufficiently large. Since here we always choose rational points for the end points of the I_{n}and since every interval with rational end points is described by two rational numbers, we see that every point x of L, that is, every real number, can be precisely described with the help of infinitely many rational numbers.

What does he mean when he says the distance between x and y would exceed the length of I_{n}? shouldn't you be able to decrease the lengths of both of them as much as you need to because there is an infinite amount of points?

What does he mean by a preassigned positive number? I'm sure that's the reason why I'm not understanding this... you have to place some sort of cap on how small you can go, right?

He says in the footnotes:

Why does this matter? Again, couldn't you just take the point on L directly in the middle of the two open endpoints and have an x in the middle of the smallest I It is important to emphasize for a nested sequence that the intervals I_{n}areclosed. If, for example, I_{n}denotes the open interval 0 < x < 1/n, then each I_{n}contains the following one and the lengths of the intervals tend to zero; but there is no x contained in all I_{n}._{n}?

Please help this is driving me crazy...

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# Help with nested intervals in Courant please

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