A linguistic reading of the definition

Remark — A linguistic reading of the definition

The definition of accumulation point is a quantified sentence: ”$\forall\,\delta>0,\ \exists$ a point of $D$ in $\dot I_\delta(x_0)$”. The universal quantifier “for every $\delta$” means that the property holds even for very small $\delta$: it is precisely this that forces the existence of infinitely many points of $D$ close to $x_0$. If we removed them one at a time, starting from the nearest, there would always remain at least one more (it suffices to choose $\delta$ smaller than the minimum distance to the nearest one remaining).

As Esty notes, in mathematics the word “every” is almost always a quantifier, not an adjective; being able to recognise this distinguishes those who truly understand the definition from those who read it merely as a sentence.

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Topics: Limits
Concepts: Accumulation point