dopeyranger said:what is the lim sup of the set containing all rationals in the closed interval [0,1] ?
and what is the lim inf?
How do I prove that the value is correct?
The lim sup (limit superior) of a sequence of rational numbers in the interval [0,1] is the largest number that can be approached by the sequence from above. Similarly, the lim inf (limit inferior) is the smallest number that can be approached by the sequence from below.
To calculate the lim sup and lim inf of a sequence of rational numbers in [0,1], we take the supremum (or least upper bound) and infimum (or greatest lower bound) of the set of all numbers that appear in the sequence. The lim sup is always equal to or greater than the lim inf.
The interval [0,1] serves as a boundary or range for the sequence of rational numbers. It restricts the values that can be approached by the sequence, making it easier to determine the lim sup and lim inf within a specific range.
The lim sup and lim inf of a sequence of rational numbers in [0,1] represent the largest and smallest values that the sequence can approach, respectively. This is similar to the concept of limits, where the limit of a sequence is the value that it approaches as the number of terms increases to infinity.
No, the lim sup and lim inf of a sequence of rational numbers in [0,1] may not always be rational numbers. It depends on the sequence and the values that it approaches. In some cases, the lim sup and lim inf may be irrational numbers or even infinity.