CSIR NET MATHEMATICS| Real Analysis | Concepts, definition and Solution of Real Analysis Dec 2018

In this video, we are going to discuss the problem of real analysis from CSIR NET December 2003. Example 73 from booklet 3, which is of mark 4.75. Let Pn denote the nth prime number when we enumerate prime numbers in the increasing order. For example, P1 equal to 2, P2 equal to 3, P3 equal to 5 and so on. Let s is equal to sn, pn plus 1 minus pn, n belongs to n and greater than equal to 1. Then, which of the following are true? The first option, supremum of s is equal to infinity, lim sup n tends to infinity, s n is equal to infinity. Third option, infimum of s is less than infinity, lim inf n tends to infinity, s n greater than equal to. Now, to solve this example, we required some concept, supremum of s, infimum of s, lim sup of sequence, and lim inf of sequence. So, first we begin with the set S. So, given set S is equal to S n P n plus 1 minus P n n belongs to n and greater than equal to n. Where P n is a prime number. P 1 equal to 2, P 2 equal to 3, P 3 equal to 4 which is in the equation order. In the number theory, this is important exact set and which is called the prime gap. So, it is a well known set which is known as prime gap. Now, the elements of S, P 1 equal to 1 and P 2 equal to 3. So, first element is P 2 minus P 1. So, 3 minus 2. Then the second element is P3 minus P2, so which is 5 minus 3. Then the third element, P4 minus P3, 7 minus 5. Fourth element, P5 minus P4 and all the other remaining elements. So, as is 1, 2, 2, 2, 4 and 10 continue. We can note one thing that 2 is only even prime number. Therefore, 3 minus 2 equal to 1 which is only the odd number in this set. First, we are going to define Supremum of S and for that we require the concept of upper bound of set S. A real number K is said to be upper bound of set S if S is less than or equal to K for all S belongs to. That means all the element of S is less than or equal to K then K is called the upper bound of set S. Supremum of a set S or least upper bound of set S. A least number of the set of upper bounds. A least number of the set of upper bound is called the supremum. For the given set, this prime gap is increased, it's never going to stop, therefore, this set does not have any upper bound, therefore, supreme of S is equal to infinity, and which is the first option, the first option is correct. Now, we are going to discuss the third option, and for that, we require the definition of lower bound of a set S. A real number K is said to be lower bound of set S if S is greater than equal to K for all S belongs to S. That means K is less than equal to S for all S belongs to A, S then K is called the lower bound of set S. Infimum of set S or greatest lower bound of set S, the greatest number of set of lower bounds of S is called the infimum. Now, for the given set, 1 is the smallest number such that all the other elements are greater than 1. So, it is satisfied by this condition S is greater than equal to 1. 1 is a lower bound of the set and all the other numbers less than 1 will also behave as a lower bound of S. Therefore, 1 is a infimum of set S and which is finite. So infimum of S is less than infinity. The third option is correct. Now, we are going to discuss the second option, which is Lim-superior. So, lim superior of a sequence. Let's define that lim sup n tends to infinity s n is equal to limit n tends to infinity supremum s m m greater than equal to m. In other words, it's infimum of supremum s m m greater than equal to n and greater than equal to 1. As we discussed, the set S does not have any upper bound and S contains the sequence s n. Therefore, s n is not bounded above sequence and therefore, supremum should be infinity and limit n tends to infinity It has to be infinity, therefore, lim sup of n tends to infinity Sn is infinity, which is the second option, which is also correct. Now, we are going to discuss the fourth option, liminf. So, limit inferior of a sequence is defined as liminf n tends to infinity Sn is equal to limit n tends to infinity infimum m greater than equal to n Sm, which is supremum of infimum Sm m greater than equal to n and greater than equal to 1. This is sequence of Sn. Now, as we discussed, 1 is only the odd number in the sequence and other numbers are the even numbers. In the even numbers, 2 is the smallest element and we know the single element cannot disturb the limiting value. Therefore, the 2 is the smallest number and in case of limit, it is behaved as a lim inf n tends to infinity Sn is always greater than equal to 2. So The fourth option, lim inf and tends to infinity as n is greater than equal to 2. So, this example has all the four options are correct. Thanks for watching. Like, share, subscribe and press bell icon of next gen.