7.4 Generalized sequences, or “nets’[29X]
[21J]Let in the following \((J,≤)\) be an ordered set with the filtering property
A function \(f:J→X\) is called net.
This \(f\) is a generalization of the concept of sequence; indeed the set \(J=ℕ\) with its usual ordering has the filtering property
In this Section we will concentrate on the case \(X=ℝ\).
[0FR] Prerequisites:3, 62, Sec. ??.
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 [ 2 ] )
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 62.
We recall from 61 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 189.
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.
[230]Having fixed \((a_ n)_{n\in {\mathbb {N}}}\) a real sequence, \((a_{n_ k})_{k\in {\mathbb {N}}}\) is a subsequence when \(n_ k\) is a strictly increasing sequence of natural numbers.
Similarly having fixed \(f:J\to {\mathbb {R}}\), let \(H\subseteq J\) be a cofinal subset (as defined in 58): We know from 4 that \(H\) is filtering. Then the restriction \(h={f}_{|{H}}\) is a net \(h:H\to {\mathbb {R}}\), and is called ”a subnet of \(f\)”.
More in general, suppose that \((H,\le _ H)\) is cofinal in \((J,\le )\) by means of a map \(i:H\to J\); this means (adapting ??) that
then \(h=f \circ i\) is a subnet.
- E239
- E239
[0FS] Prerequisites:237,234,5.Show that if the limit \(\lim _{j∈ J}f(j)\) exists, then it is unique.
- E239
[0FT]Suppose \(f\) is monotonic, show that \(\lim _{j∈ J} f(j)\) exists (possibly infinite) and coincides with \(\sup _ J f\) (if it is increasing) or with \(\inf _ J f\) (if it is decreasing).
Infer that
\begin{eqnarray*} \limsup _{j∈ J}f(j){\stackrel{.}{=}}\lim _{j∈ J} \sup _{k≥ j} f(k)\\ \liminf _{j∈ J}f(j){\stackrel{.}{=}}\lim _{j∈ J} \inf _{k≥ j} f(k) \end{eqnarray*}are always well defined.
- E239
[0FV]Show that the limit exists \(\lim _{j∈ J}f(j)=ℓ∈\overlineℝ\) if and only if
\[ \limsup _{j∈ J}f(j)=\liminf _{j∈ J}f(j)=ℓ~ . \]- E239
[22Y] Prerequisites:53,4,237,234,6.Suppose \(H⊆ J\) is cofinal and let \(h={f}_{|{H}}\) be the subnet (as defined in 238);
Suppose that \(\lim _{j∈ J}f(j)=l∈ \overlineℝ\) show that \(\lim _{j∈ H}h(j)=l\).
Similarly if \((H,\le _ H)\) is cofinal in \((J,\le )\) by means of a map \(i:H\to J\), and \(h=f \circ i\).
[237]Suppose that the set \(J\) is directed but not filtering; then by 3 it admits a maximum element that we call \(\infty \); the above definitions and properties can also be stated in this case, but they are trivial, since