show directly from the definition that the following are not cauchy sequences. (

Question

show directly from the definition that the following are not cauchy sequences.

(a) ((-1)n)

c) (ln n)

Explanation / Answer

So the definition of a Cauchy sequence {a_i} is one where given e > 0 there exists a positive integer N such that for every r,s > N, |a_r - a_s| 0 be given, we want to show that no suitable N can be found that fulfills the above definition of Cauchy sequences. Suppose on the contrary that such an N can be found. Let s = 2N and r = 2M. Then |a_r - a_s| = |(2M+(-1)^{2M} / 2M) - (2N+(-1)^{2N} / 2N)| = |(2M + 1/(2M)) - (2N + 1/(2N))| >= |2M - 2N - 1| Since e is fixed, we can choose M large enough so that |2M - 2N - 1| > e i.e. M > (e+1)/2 + N. Hence we have |a_r - a_s| > e for every possible choice of N, thus showing that the conditions of a Cauchy sequence cannot be satisfied. hope helps

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