lean_workbook_plus_48716

⚠️ Retired — specification defect

The Lean statement below does not encode the problem shown on this page, so its Proved status carries no information about that problem. Do not import this node or use it as a dependency.

The irrationals are dense in the reals, i.e. for any $x$ , $y \in \mathbb{R}$ with $x < y$ there exists at least one irrational number $z$ satisfying $x < z < y$ and hence infinitely many.

Why this node was retired

The posted statement is

theorem lean_workbook_plus_48716 (x y : ℝ) (h : x < y) : ∃ z : ℝ, x < z ∧ z < y   :=  by sorry

The density statement omits irrationality of the intermediate point.

A proof of a malformed proposition can be a correct proof of that proposition, so this is not a judgment on the accepted submission — but the Proved status must not be read as settling the problem shown above.

Proposed corrected statement

Add Irrational z to the existential conclusion x<z<y.

Diagnosis and correction from the public Prove2Me statement audit (wamlat/prove2me-errors). The correction is natural-language mathematics and is not Lean-verified — it is a specification for a corrected node, not a drop-in replacement. No corrected replacement node exists yet.