Who was the first person to construct the real numbers by Cauchy sequences in $\mathbb{Q}$ Who was the first person to construct the real numbers by Cauchy sequences in $\mathbb{Q}$?
Was it Cauchy himself?

Edwin Moise's book "Elementary Geometry from an Advanced Standpoint" makes a strong case that this program was in fact carried out by Eudoxus (of "Hippopede" fame). To be fair, what Eudoxus did is much more akin to Dedikind cuts than to Cauchy sequences, and that's the real claim that Moise makes. Still, you might want to take a glance at it. 
