Assume a real-valued net has a cluster point $a\in \mathbb R$. Is there a cofinal subnet converging to $a$? Let $(D, \geq)$ be a directed set and let $(n_d)_{d\in D}$ be a real-valued net. Assume $a\in \mathbb R$ is a cluster point of $(n_d)$, i.e., for every neighborhood $U$ of $a$ and every $d\in D$ there exists $d'\geq d$ such that $n_{d'}\in U$.
It is a standard fact that there exists a subnet of $(n_d)$ that converges to $a$.
But is it possible to find a cofinal subnet (see for example wikipedia) with the same property? To rephrase: is there a cofinal subset $D'\subset D$ such that $\lim_{d\in D'}n_d = a$?
If $(n_d)$ takes value in an arbitrary topological space $X$, then this is not true. See for example this answer and this one. My question is specific to the case $X=\mathbb R$, and for this space I could not find any counter-example nor proof.

Not necessarily.
Let $D=\omega_1\times\omega$ ordered lexicographically: $\langle\alpha,m\rangle\preceq\langle\beta,n\rangle$ iff $\alpha<\beta$, or $\alpha=\beta$ and $m\le n$; this is clearly a directed set. Define a net
$$\nu:D\to\Bbb R:\langle\alpha,n\rangle\mapsto 2^{-n}\,;$$
clearly $0$ is a cluster point of $\nu$.
Let $C$ be a cofinal subset of $D$. For each $n\in\omega$ let $C_n=C\cap(\omega_1\times\{n\})$; $|C|=\omega_1$, so there is some $m\in\omega$ such that $|C_m|=\omega_1$. Then $C_m$ is cofinal in $C$, but $\nu[C_m]=\{2^{-m}\}$ is disjoint from the nbhd $(-2^{-m},2^{-m})$ of $0$. Thus, no cofinal subnet of $\nu$ converges to $0$.
