Example of Convergent Sequence in R2 | L5 | TYBSc Maths | Completeness @ranjankhatu

hi everyone in this video we are going to discuss this example so we have a sequence x n it is a sequence in R2 and we have to prove that it is convergent in R2 with a euclidean distance right but the problem is that that sequence x n is defined in two different ways ok so ah first nine terms is defined using this Formula First component is 2 raised to n and the second is one by an d all remaining terms are defined in this way first component is 2 raised to 10 and the second is minus 1 by so we had to prove that it is convergent so let me mention we will prove that we will prove that that sequence x n converges 2 okay so we have to find one point where the sigma x n converges right ah first nine four first nine terms x n is defined in this way that means first nine terms are defined using this formula and remaining infinite limited terms are defined using second formula so we know that the limit of sequence is not decided by first finite terms actually it is decided by the all the remaining infinitely many terms that is why we will not give much importance to this definition our mainly focus on second definition getting since it is true for infinitely many numbers that means for all n greater than or equal to 10. let us think about the first component 2 raise to 10 it's a constant if you take any limit you will get the same so that's why the first component will be ah limit of 2 raised to 10 is 2 raise to 10 since it is constant let us think about the second component minus one by n if you apply the limit n tends to Infinity its value will be 0 so that means the second component Converses to 0 so I will take 0 here so now our Target is to prove the given sequence x n converges to this point two raised to 10 comma 0 right uh I'm going to prove this thing using let us take Epsilon let Epsilon greater than 0 B given I am sure you are familiar with the definition then also I will write the definition here so you can easily see and ah guess the my next steps so distance between x n and X should be less than Epsilon for all n greater than or equal to capital N actually our main task is to find this capital in for which this definition will be satisfied