CSIR NET MATHEMATICS | Real Analysis | Convergence of Sequences Q.65(A) Q.74(B) Q.62(C)

Hello friends, in this video we are going to discuss problem of real analysis from CSI NIT June 2019. Example no. 62 from booklet C marks 4.75 shows that xn is a sequence of positive reals. Let yn equal to xn divided by 1 plus xn then which of the following are true? option xn is convergent if yn is convergent second option yn is convergent if xn is convergent third option yn is bounded if xn is bounded and the fourth option xn is bounded if yn is bounded now we have given xn is a sequence of positive here now we have one result of sequence limits of convergent sequence preserves the inequality that means yn less than equal to xn for all n implies limit n tends to infinity y n less than equal to limit n tends to infinity x n since x n is a sequence of positive reals therefore 1 less than equal to 1 plus x n because it is a positive real number so we add x n in 1 then it will bigger than the 1 so you can write down 1 divided by 1 plus x n less than 1 multiply both side by x n then we have x n divided by 1 plus x n less than xn but this is a form of yn so we can say that yn less than xn therefore if xn less than equal to m then yn must be less than equal to m so by the definition of bounded sequence if xn is bounded then yn must be bounded so which is option number three it is correct and next if limit n n tends to infinity xn is equal to l that means xn is convergent can converge to l then limit n tends to infinity yn is equal to limit n tends to infinity xn divided by 1 plus xn is equal to limit n tends to infinity xn divided by 1 plus limit n tends to infinity xn which is convergent so l divided by 1 plus l so if xn is convergent to l then yn converged to l upon 1 plus l therefore if xn is convergent then yn is convergent so option number two is correct yn equal to xn divided by 1 plus xn implies 1 plus xn equal to xn now if you open the bracket then we have yn plus yn into xn equal to xn and if you make xn as a subject of the equation then xn equal to yn upon 1 minus yn so if we take yn is equal to constant sequence 1 then yn is bounded by 1 but xn cannot be bounded if put yn equal to 1 then xn will be infinity so yn may be bounded but xn may not be and the fourth option which says yn is bounded then xn is bounded which is wrong and similarly limit n tends to infinity yn equal to 1 then xn is equal to 1 upon 1 minus 1 so xn converges to infinity yn is convergent sequence but xn is not convergent sequence so option number one is wrong so this example has only two correct answer second and third thanks for watching for more videos like share subscribe press the bell icon of next gen