Let ${a_n}$ and ${b_n}$ be two sequences with following property: $\forall \epsilon >0,\ \exists n_0\in \mathbb{N}$ such that $n>n_0 \implies |a_n-b_n| <\epsilon$.

How do I show that $$\lim_{n \to \infty}a_n=L \iff \lim_{n\to \infty}b_n=L?$$

I tried to solve this problem by cauchy sequence but I unable to prove that this is my attempt


$|b_m-b_n |<e/2----(2)$

but I can't go further.


1 Answer 1


hint: $|b_n-L| \leq |b_n-a_n|+|a_n-L|$, and $|a_n-L| \leq |a_n-b_n|+|b_n-L|$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.