EDB — 20C

[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 \(-∞\)