Let {An} and {Bn} be two sequences and suppose that the set {n:Anis not equal to Bn} is finite.?

Question

Prove that the sequences either both converge to the same limit or both diverge?

Answer 1

Since the set {n:Anis not equal to Bn} is a finite subset of the postive integers, it has a maximum element k. So, for n >= k+1 we have A(n) = B(n), that is, the k +1 tails of A(n) and B(n) are exactly the same sequence.

If A(n) converges to some A, then its k+1 tail converges to a and so does the k +1 tail of B(n). Therefore, B(n)also converges to A.

If A(n) diverges, so does its k+1 tail, so that the k+1 tail of f B(n) diverges. Threfore, B(n) diverges.

This is consequence of the fact that convergence or divergenc of a sequence depends on its ultimate behavior, that is, deleting a finire number of terms does'nt affect convergence or divergence or its limit in case of convergence.