convergence divergence - Cauchy sequences of real numbers, can we construct "new numbers?"

We can construct reals by defining a real number as a equivalence class of rational Cauchy sequences. The set of all such equivalence classes have a cardinality strictly larger than $\mathbb Q$.

Now consider, in the same fashion, Cauchy sequences of real numbers. Consider two such sequences $\left\{a_k\right\}_{k=1}^\infty$ and $\left\{b_k\right\}_{k=1}^\infty$. Define an equivalence relation, in the same way, by $$(a\sim b) \iff \displaystyle\lim_{k\to\infty} \left|a_k - b_k\right| =0$$

By similar proofs as in the rational case, this relation is really an equivalence relation. Now, consider the set of all equivalence classes with respect to the relation $\sim$ defined above. Will this give us a "new type of numbers" or will we just get $\mathbb R$ all over again?

If the cardinality of the set of all equivalence classes of real Cauchy sequences is $|\mathbb R|$, how would one go on and prove it?

Answer

Each Cauchy sequence $(a_n)$ of reals tends to a real limit $a^*$. Under
your definition $(a_n)\sim (b_n)$ iff $a^*=b^*$. So, you'll just get the
real numbers again.