Nets and sequences in a 1st couuntable space.

Question

Let $(X,\mathcal{T}$) be a 1st countable topological space. Let $(x_\delta)_{\delta\in\Delta}$ be a net converging to $x$. Does there exist a sequence $(x_n)$ that converges to $x$ and which is a subnet of $(x_\delta)$? Any feedback is most appreciated?

Answer 1

Certainly not. Even if the net $(x_\delta)$ has the amazing property $x_\delta = x$ for all $\delta$. Even then, it may have no subsequences at all.