negation of a quantifier

Negating a quantified statement follows two simple, mechanical rules. The negation of “for all x, P(x)” is “there exists x such that not P(x)”; the negation of “there exists x such that P(x)” is “for all x, not P(x)”. In words: to deny that something holds for everything, you only need one exception; to deny that something exists, you must rule it out everywhere.

These rules let you push a negation inward through a whole stack of quantifiers: each ∀ flips to ∃, each ∃ flips to ∀, and the innermost predicate gets negated last. For nested statements you apply them in sequence, also using De Morgan's laws on the connectives along the way. The mechanical reliability of this process is precisely why mathematicians prize the quantifier notation.

This is the practical engine behind every “suppose not” argument in analysis. To show a sequence does NOT converge to L, you negate “for all epsilon > 0 there exists N such that for all n ≥ N, |a_n − L| < epsilon”, obtaining “there exists epsilon > 0 such that for all N there exists n ≥ N with |a_n − L| ≥ epsilon”. The negated form tells you exactly what to produce: a stubborn epsilon and, for each N, a late index that still misbehaves.