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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

$|a_m-a_n|<e/2----(1)$

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

but I can't go further.

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

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