PLEASE DO IT BY USING THE HINT,THANKS. Consider the following sequence: As we progress into this sequence we encounter more and more zeroes while 1s get sparser. Prove that this sequence does NOT converge to 0. Hint: try to determine the values of epsilon > 0 for which it possible to find a suitable N N, and those epsilon > 0 for which no such N exists. Solution If the sequence converges to 0, then for = 1/2, there exists an N such that for all n >= N, |x n - 0| < 1/2 Note that the k(k+1)/2 term = 1 for all k Let N be the value such that for all n >= N, |x n - 0| < 1/2 Let Z be the smallest integer >= N Z(Z+1)/2 > N for all N >= 2 Yet, x Z(Z+1)/2 = 1, so |x Z(Z+1)/2 - 0| = 1 > 1/2 = This is a contradiction. Thus, the sequence does not converge to 0. .

1. PLEASE DO IT BY USING THE HINT,THANKS.
Consider the following sequence: As we progress into this sequence we encounter more and
more zeroes while 1s get sparser. Prove that this sequence does NOT converge to 0. Hint: try to
determine the values of epsilon > 0 for which it possible to find a suitable N N, and those epsilon
> 0 for which no such N exists.
Solution
If the sequence converges to 0, then for = 1/2, there exists an N such that for all n >= N,
|x n - 0| < 1/2
Note that the k(k+1)/2 term = 1 for all k
Let N be the value such that for all n >= N, |x n - 0| < 1/2
Let Z be the smallest integer >= N
Z(Z+1)/2 > N for all N >= 2
Yet, x Z(Z+1)/2 = 1, so |x Z(Z+1)/2 - 0| = 1 > 1/2 =
This is a contradiction. Thus, the sequence does not converge to 0.