Suppose a sequence aj has the property that for every natual

Suppose a sequence {a_j} has the property that, for every natual number N, ther is a j_N such that a_jn = a_{jn +1}= ....= a_jN+N. In other words, the sequence has arbitrarily long repetitive strings. Does it follow that the sequence converges? Can you prove your answer?

Solution

The sequence may converge or diverge.Because for example if we take any convergent or divergent sequence then if we add the repetitive string in that sequence..It will not change the convergence or divergence of that original sequence.