Here, $\mathbb{N}$ designates the unordered and undirected set of nonnegative integers. Let $A=\mathbb{N}$ be directed by $\leq$ and let $B=\mathbb{N}$ be directed by the relation $\preccurlyeq$ that coincides with $\leq$ on $\{2n\}_{n\in\mathbb{N}}$ and on $\{2n+1\}_{n\in\mathbb{N}}$, and satisfies $\left(\forall m\in\mathbb{N}\right)\left(\forall n\in\mathbb{N}\right)2m+1\preccurlyeq 2n$. Now let $\left(x_a\right)_{a\in A}$ be a sequence in $\mathbb{R}$.
I am asked to first show that $\forall a\in A$, $\forall b\in B$, $2a\preccurlyeq b\implies a\leq 3b$ and then to show that $\left( x_{3b}\right)_{b\in B}$ is a subnet but not a subsequence.
For the first part, I have the following: Let $a\in A$ and $b\in B$ such that $2a\preccurlyeq b$. Since the left hand side is even, the right hand side must also be even by definition of $\preccurlyeq$, i.e. $b=2n$ for some $n\in\mathbb{N}$. Then we have $2a\preccurlyeq 2n\implies 2a\leq 2n\implies a\leq n\implies a\leq 6n = 3b$. Is this sufficient or am I missing some details, it seems a bit easy?
For the second part, I have this: The net $\left(y_b\right)_{b\in B}$ is a subnet of the net $\left( x_a\right)_{a\in A}$ via $k:B\rightarrow A$ iff $\forall b\in B, y_b = x_{k(b)}$ and $\forall a\in A, \exists d\in B$ such that $\forall b\in B$, $d\preccurlyeq b\implies a\preccurlyeq k(b)$. Now obviously the first part is satisfied with $k(b) = 3b$. The next is to show the second property, so let $a\in A$ and choose $d=a$.
If $a$ is even, then $\forall b\in B$ with $d\preccurlyeq b$ we have that $b$ is even as well by definition of $\preccurlyeq$. Then, $d\preccurlyeq b\implies a\leq b\implies a\preccurlyeq 3b$ since $3b$ is also even and thus $a\leq 3b$.
If $a$ is odd, then $d\preccurlyeq b\implies \begin{cases} a\leq b\implies a\leq 3b & \mbox{if }b\mbox{ is odd}\\ a\preccurlyeq 3b & \mbox{if }b\mbox{ is even}\end{cases}$. I think this shows it's a subnet?
Finally, I am stuck on what to do for showing it is not a subsequence. From what I've read there are two paths to show something is not a subsequence of a net, either you show that $B$ is not countable or that $k$ is not strictly increasing. Since $B=\mathbb{N}$ here, we must try to show that $k$ is not strictly increasing? I am confused because here we have two order relations but in the definition given we have only $\preccurlyeq$ for both order relations.