Prove that a sequence cannot converge to two different limits.
Proof. Consider a sequence such that
We show that we must have . By the definition of the limit we know that for all there exist a positive integers and such that
Let , then for all and all we have
(The final line follows since if , then , so setting would contradict that for all .) Therefore, a sequence cannot converge to two different limits
You changed the sign in front of a_n when you applied the triangle inequality. How is that possible?
The absolute value of a number is equal to the absolute value of its negative