Prove or disprove the inequality with a quantifier

∃x∈R,∀y∈R,x−y2<0\exists x \in \mathbb{R}, \forall y \in \mathbb{R}, x - y^2 < 0 Let x=y2−1x = y^2 - 1 then y2−y2−1=−1<0y^2 - y^2 - 1 = -1 < 0 This proof got marked wrong, why? =================      If x=−5x=-5 then what would you pick for yy? – imranfat 2 days ago      Say it aloud. There exists an xx, for all y,x−y2<0y, x-y^2< 0. You must pick your xx first, and not let your xx depend on y.y. If it was phrased ∀y,∃x\forall y, \exists x that would mean something else, and you would be aloud to proceed as you did. – Doug M 2 days ago ================= 1 Answer 1 ================= The existential quantifier comes first in the statement, meaning that the chosen xx must work for all yy. You defined xx in terms of yy. A correct proof would be: Let x=−1x=-1. Then, for all y∈Ry\in\mathbb{R}, x=−1<0≤y2⟹x−y2<0x=-1<0\leq y^{2}\implies x-y^{2}<0. What you proved is essentially that ∀y∈R,∃x∈R:x−y2<0. \forall y\in\mathbb{R},\exists x\in\mathbb{R}:x-y^{2}<0.