If $X$ is first-countable then a net converges when a subsequence converges?

Question

Let be $X$ and we assume that $(x_\lambda)_{\lambda\in\Lambda}$ is a net such that there exists a cofinal and increasing map $\varphi$ form $\Bbb N$ to $\Lambda$ such that $\big(x_{\varphi(n)}\big)_{n\in\Bbb N}$ converges to any $x_0\in X$: so if $X$ is first countable then the convergence of $\big(x_{\varphi(n)}\big)_{n\in\Bbb N}$ to $x_0$ implies the convergence of $(x_\lambda)_{\lambda\in\Lambda}$ to $x_0$? If the answer to the last question is negative then there is a relevant reason different form this or this or rather this that show why into first countable space is sufficent to consider only sequence?

Answer 1

As Ruy suggested above into the comments the conjecture is generally false: e.g. the sequence $(x_n)_{n\in\Bbb N}$ defined as $$ x_n:=\begin{cases}0,\,\text{if }n\,\text{is even}\\n,\,\text{if }n\,\text{is even}\end{cases} $$ is a sequence into a first countable space that does not converege but it has trivially a converging subsequence.