Linguistic reading of the quantifiers

The definition of a limit is an exercise in logic: three quantifiers in a precise order, which cannot be changed.

Observation — Linguistic reading: $\forall\varepsilon\,\exists n_0\,\forall n\ge n_0$

The definition of a limit has three nested quantifiers in the order "$\forall\varepsilon\,\exists n_0\,\forall n\ge n_0$". The order cannot be swapped: "$\exists n_0\,\forall\varepsilon$" would be false (no single $n_0$ suffices for all $\varepsilon$). The value $n_0$ depends on $\varepsilon$: the smaller $\varepsilon$ is, the larger (in general) $n_0$ will be.

The negation of "$a_n\to L$" is therefore ”$\exists\varepsilon>0\,\forall n_0\,\exists n\ge n_0$ such that $|a_n-L|\ge\varepsilon$”: there exists a level $\varepsilon$ that the sequence keeps exceeding for arbitrarily large indices (Esty).

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Topics: Sequences
Concepts: Limit of a sequence · Quantifiers