[0FR] Prerequisites:[06V], [06Y], Sec. [1YY].
Given \(J\) a (possibly partially) ordered and filtering set, and given \(f:J→ℝ\), we want to define the concept of limit of \(f(j)\) ”for \(j→∞\)”. 1 .
We will say that
\[ \lim _{j∈ J}f(j)=l∈ ℝ \]if
\[ ∀ \varepsilon {\gt}0 \, ∃ k∈ J \, ∀ j∈ J, \, j≥ k ⇒ |l-f(j)|{\lt}\varepsilon \quad . \]Similarly limits are defined \(l=±∞\) (imitating the definitions used when \(J=ℕ\).) (This is the definition in the course notes, chap. 4 sect. 2 in [ 3 ] )
Equivalently we can say that
\[ \lim _{j∈ J}f(j)=l∈ \overlineℝ \]if for every neighborhood \(U\) of \(l\) we have that \(f(j)∈ U\) eventually for \(j∈ J\); where eventually has been defined in [06Y].
We recall from [231] that ”a neighborhood of \(∞\) in \(J\)” is a subset \(U⊆ J\) such that \(∃ k∈ J ∀ j∈ J , j≥ k ⇒ j∈ U\). Then we can imitate the definition [20D].
Fixed \(l∈\overline{ℝ}\) we have \(\lim _{j∈ J}f(j)=l\) when for every ”full” neighborhood \(V\) of \(l\) in \(ℝ\), there exists a neighborhood \(U\) of \(∞\) in \(J\) such that \(f(U)⊆ V\).
In particular, this last definition can be used to define the limits of \(f:J→ E\) where \(E\) is a topological space.