6.2 Frequently, eventually[29K]
We will write \(\overline{ℝ}\) for \(ℝ∪\{ ±∞\} \).
[0BG] Given \(A⊆ ℝ\), a point \(x∈ \overline{ℝ}\) is called accumulation point for \(A\) if every deleted neighborhood of \(x\) intersects \(A\).
[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”.
[22X]If \(x_ 0∈\overlineℝ\) is not an accumulation point for \(I\), then we always have that ”\(P(x)\) definitely is true, for \(x\) tending to \(x_ 0\)” .
[20C]Suppose for simplicity that \(I=ℝ\). Putting together the previous ideas, we can write equivalently:
if \(x_ 0∈ℝ\),
\(∃ 𝛿 {\gt}0, ∀ x≠ x_ 0, |x-x_ 0|{\lt}𝛿⇒ P(x)\)
\(P(x)\) definitely applies for \(x\) tending to \(x_ 0\)
\(∀ 𝛿 {\gt}0, ∃ x≠ x_ 0, |x-x_ 0|{\lt}𝛿∧ P(x)\)
\(P(x)\) frequently applies for \(x\) tending to \(x_ 0\)
whereas in case \(x_ 0=∞\)
\(∃ y∈ℝ, ∀ x, x{\gt}y⇒ P(x)\)
\(P(x)\) definitely applies for \(x\) tending to \(∞\)
\(∀ y∈ℝ, ∃ x, x{\gt}y ∧ P(x)\)
\(P(x)\) frequently applies for \(x\) tending to \(∞\)
and similarly \(x_ 0=-∞\)
\(∃ y∈ℝ, ∀ x, x{\lt}y⇒ P(x)\)
\(P(x)\) definitely applies for \(x\) tending to \(-∞\)
\(∀ y∈ℝ, ∃ x, x{\lt}y∧ P(x)\)
\(P(x)\) frequently applies for \(x\) tending to \(-∞\)