Definition
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[0B3] Let \(I⊆ ℝ\) be a set, \(x_ 0∈\overlineℝ\) an accumulation point for \(I\). Let P(x) be a logical proposition that we can evaluate for \(x∈ I\). We define that
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”P(x) holds eventually for \(x\) tending to \(x_ 0\)” if |
there is a neighborhood \(U\) of \(x_ 0\) \(∀ x∈ U∩ I, \) P(x) is true ; |
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”P(x) frequently holds for \(x\) tending to \(x_ 0\)” if |
for every neighborhood \(U\) of \(x_ 0\) \(∃ x∈ U∩ I\) for which P(x) ; |
where it is meant that the neighbourhoods are ”deleted”.