Here are basically two questions. The first, what is the cardinal of equivalent Cauchy sequences of rationals? I know it's $\beth_1$ because of the set is essentially real numbers. But I want to know how to obtain this in view of Cauchy sequences, instead of decimal expansions. I mean, how to construct a one-to-one mapping from equivalent sequences to subsets of integers? Furthermore, what's the cardinal of Cauchy sequences? Is that ${\beth_1}^{\beth_0}=\beth_1$?
Edit
Thanks all commentators for indicating the error sourced from absence of (generalized) continuum hypothesis. Now my problem becomes asking for an explicit mapping from set of equivalent Cauchy sequences to power set of integers without computation of cardinals.
The set of all sequences rational numbers is $\mathbb{Q}^\mathbb{N}$, and it's cardinality is $$|\mathbb{Q}^\mathbb{N}|=|\mathbb{N}^\mathbb{N}|\leq|\mathscr{P}(\mathbb{N})^\mathbb{N}|=|\mathscr{P}(\mathbb{N}\times\mathbb{N})|=|\mathscr{P}(\mathbb{N})|,$$ where $\mathscr{P}(\mathbb{N})$ is the power set of $\mathbb{N}$. In other words:
Take two injections $\phi:\mathbb{Q}\rightarrow\mathbb{N}$ and $\psi:\mathbb{N}\times\mathbb{N}\rightarrow\mathbb{N}$ (you can construct these explicitly, using the well-ordering of $\mathbb{N}$). Then the function $\varphi:\mathbb{Q}^\mathbb{N}\rightarrow\mathscr{P}(\mathbb{N})$ given by $$\varphi(f)=\psi\left(\left\{(n,\phi(f(n))):n\in\mathbb{N}\right\}\right)$$ is injective, hence so is its restricition to the Cauchy sequences.