There’s two common ways to construct the real number system: Dedekind cuts and equivalence classes of Cauchy sequences. I’d like to try a different way.
Let $X$ be the set of all functions $f:Q\rightarrow Q$ such that for any positive rational number $\epsilon$ there exists a positive rational number $\delta$ such that for all rational numbers $x,y>\delta$, we have $|f(x)-f(y)|<\epsilon$. (Intuitively, it means that its limit as $x$ goes $\infty$ is a real number.) And let’s define an equivalence relation $\sim$ on $X$ by saying that $f\sim g$ if the limit of $f(x)-g(x)$ as $x\rightarrow\infty$ is $0$.
My question is, is the set of equivalence classes of elements of $X$ under $\sim$ isomorphic to the set of real numbers?