Definition
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[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 [06P]): We know from [06X] that \(H\) is filtering. Then the restriction \(h={f}_{|{H}}\) is a net \(h:H\to {\mathbb {R}}\), and is called βa subnet of \(f\)β.
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 [(3.121)]) that
\begin{equation} \label{eq:cofinale_ H,J} ( β h_ 1,h_ 2β H, h_ 1β€_ H h_ 2β i(h_ 1)β€ i(h_ 2) ) ~ β§~ (β jβ J ~ β hβ H,~ i(h)β₯ j) \quad ; \end{equation}
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then \(h=f \circ i\) is a subnet.