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

Hence,

(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?

(Kindly asking)

The absolute value of a number is equal to the absolute value of its negative