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**lllll** Show that if $\displaystyle \{a_n\}$ and $\displaystyle \{b_n\}$ are Cauchy in $\displaystyle \mathbb{Q}$ contained in equivalence classes $\displaystyle a$ and $\displaystyle b$ (real numbers) respectively, and if $\displaystyle a_n < b_n$ for all $\displaystyle n$, then there exists a real number $\displaystyle \delta$ which contains a rational Cauchy sequence $\displaystyle \{\delta_n\}$ each of whose $\displaystyle \delta_n$ is positive and $\displaystyle b=a+\delta$.

Here's what I have, for some reason it doesn't seem correct.

since $\displaystyle a_n < b_n$ then $\displaystyle 0 < b_n-a_n$ implying that there exist $\displaystyle \delta_n$ such that $\displaystyle \delta_n = b_n-a_n$ for all $\displaystyle n$