# Prove Sequence is Converges or diverge

• Feb 17th 2010, 04:39 PM
summerset353
Prove Sequence is Converges or diverge
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
• Feb 17th 2010, 07:26 PM
Drexel28
Quote:

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