@XanderHenderson: To sum up the key points: (1)
ℚ is an ordered field containing ℤ, and only the axioms [ℤ ⊆ ℚ] and [ℚ = ℤ/ℤ*] distinguish ℚ from other ordered fields, and from the other axioms you can prove ∀a,b∈ℚ ( ∀c∈ℚ ( c > 0 ∧ a ≤ b+c ) ⇒ a ≤ b ), so this basic real analysis fact actually holds for all ordered fields; (2) You can prove this fact intuitionistically, because ∨elim and explosion are intuitionistic.