Prove Sequence is Converges or diverge

**summerset353** · February 17th 2010, 04:39 PM

Let {a_n} and {b_n} be two sequences and suppose that the set {n: (a_n) does not equal (b_n) is finite. Prove the sequences either both converge to the same limit or both diverge.

I know the set is finite, so it contains a largest number N. How do the sequences compare for all n>N and prove

**Drexel28** · February 17th 2010, 07:26 PM

Originally Posted by summerset353

Let {a_n} and {b_n} be two sequences and suppose that the set {n: (a_n) does not equal (b_n) is finite. Prove the sequences either both converge to the same limit or both diverge.

I know the set is finite, so it contains a largest number N. How do the sequences compare for all n>N and prove

Let $n_1,\cdots,n_m$ be the finite set of natural numbers such that $a_{n_k}\ne b_{n_k}$ then we have that $N\leqslant n\implies a_n=b_n$ where $N=\max_{1\leqslant \ell\leqslant m}n_\ell$ . So...

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