Comment by Gro-Tsen on Equivalence of omniscience principles for natural numbers and analytic omniscience principles for Cauchy real numbers

By a “Cauchy sequence” $(u_n)$, do you mean a Cauchy-with-modulus sequence $∃μ:\mathbb{N}\to\mathbb{N}.∀k.∀p,q≥μ(k).|u_p-u_q|<2^{-k}$ or a Cauchy-without-modulus sequence $∀k.∃n.∀p,q≥n.|u_p-u_q|<2^{-k}$ ? Both have been called “Cauchy sequence”, so it would really be best to avoid the term without clarifying it as “with modulus” or “without modulus”.