I was wrapping my head around the difference between compactness and sequential compactness, and more specifically why compactness does not necessarily imply sequential compactness. At the root, I am missing the intuition as to why a convergent subnet of a subsequence does not necessarily allow us to retrieve a convergent subsequence.
We can talk about convergence, but I think the matter lies at an even earlier stage than that. Take $X$ a set (not necessarily a topological space) and a subset $S$. Consider a sequence $(x_n)_{n\in\mathbb N}$ over $X$ and a subnet $x_{\alpha(\bullet)}$ that lies eventually in $S$. Here $\alpha\colon (A,\leq) \to (\mathbb N,\leq)$ is a monotone final function, and the previous property simply means that there exists $a\in A$ such that $x_{\alpha(A_{\geq a})}\in S$, where $A_{\geq a} \triangleq \{b\in A, ~b\geq a\}$. Naively, one could think that this subnet (which is not necessarily a sequence) should yield a subsequence eventually in $S$ after "extracting the relevant terms." More specifically, $\alpha(A) \subset \mathbb N$ being a cofinal subset, it seems natural to consider the monotone final function $\beta\colon\mathbb N\to\mathbb N$ which indexes the elements of $\alpha(A)$ in order. Naturally, $x_{\beta(\bullet)} = (x_{\beta(n)})_{n\in\mathbb N}$ is a subsequence of $x$, the issue is that it may not be in $S$ eventually. Indeed, the set $\alpha(A_{\geq a})$ is a subset of $\alpha(A)$, which is not necessarily cofinal (and here lies the crux I believe), so that it may not contain $\beta(\mathbb N_{\geq k})$, and thus we cannot conclude that $x_{\beta(\bullet)}$ eventually is in $S$.
Essentially, I would like to gain an intuition as to what "prevents" the extraction of a subsequence eventually in $S$ from a subnet eventually in $S$ of a sequence. A simple example of a sequence with a converging subnet but no converging subsequence would be equally appreciated, or even without convergence simply using the notion of being eventually in a set. I have found a neat example here, a sequence in a compact space without converging subsequence is given, but I fail to see what would be a convergent subnet and what really prevents extracting a subsequence. I have also found an example here where the subnet is explicitly given, but I find the Čech-Stone compactification a bit too involved to gain an intuition.
