The following is a definition of subnet using a function between two directed sets: (cropped scan)
Def. Let $P: \Lambda \rightarrow X$ and $Q: M \rightarrow X$ be nets (where $\Lambda$ and $M$ are directed sets) The net $Q$ is called a subnet of the net $P$, if there exists a function $\varphi: M \rightarrow \Lambda$ such that:
- $Q=P \circ \varphi$
- $\left(\forall \lambda_{0} \in \Lambda\right)\left(\exists \mu_{0} \in M\right),(\forall \mu \in M)$ if $\mu \geqslant \mu_{0}$ then $\varphi(\mu) \geqslant \lambda_{0}$.
We choose this definition of the function $\varphi$ instead of just using a strictly increasing function in the "Sequence and subsequence" situation, to make sure we have the property $2)$ listed here. In the "Sequence and subsequence" situation, in essence we are using a strictly increasing function $f$ from $\mathbb{N}$ to $\mathbb{N}$ to construct a subsequence.
My question is: why a strictly increasing function $\varphi$ doesn't guarantee property $2)$? I need to find a simple counterexample but I can't.
By "Strictly increasing", I mean:
If $\mu, \mu' \in M$, $\mu' > \mu$ (which means $\mu' \geq \mu$, but $\mu' \neq \mu$), we have $\varphi(\mu') > \varphi(\mu)$, (which means $\varphi(\mu') \geq \varphi(\mu)$, but $\varphi(\mu') \neq \varphi(\mu)$)
Property 2) describes what is called "cofinality" of the subset $\varphi(M) \subset \Lambda$. As an example take $M = \mathbb N$ and $\Lambda = \mathbb R $ with their usual order. Then $\varphi(n) = \sum_{i=1}^n 2^{-i}$ is strictly increasing, but all $\varphi(n) < 1$.