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?
 A: 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. 
A: A construction of this type (though not exactly the same) is usually attributed to Cantor in 1872.  However, historians who have looked into this matter in detail note that Meray presented such a construction earlier, namely in 1869.
M´eray, H. C. R.: Remarques sur la nature des quantit´es d´efinies par la condition
de servir de limites `a des variables donn´ees, Revue des soci´eti´es savantes
des d´epartments, Section sciences math´ematiques, physiques et naturelles, 4th
ser., 10 (1869), 280–289.
For a discussion, see this article.
A: As John said, this idea goes back to Eudoxus.
In particular, in order for Eudoxus to show that Thales' Theorem, he first showed it when the ratio of the segments was a rational number, and then showed the general case by approximating an arbitrary ratio of segments by rational numbers. He used the same idea in order  to show that the ratio of the perimeters of circles is equal to the ratio of their radii by approximating the perimeters of circles by perimeters of inscribed canonical polygons.
