I have a set $ B $ that can be written as $ B = \cup_{\nu < \lambda} B_{\nu} $ where $ \kappa $ is the cardinality of $ B $, that is uncountable, and $ \vert B_{\nu} \vert < \kappa $ with $ \lambda $ being the cofinality of $ B $.
The problem is that I want the sequence $ \{ B_{\nu} : \nu < \lambda \} $ to be an ascending sequence. In other words, I want $ B_{\nu} \subseteq B_{\mu} $ for all $ \nu < \mu < \lambda $. How is that possible?
Intuitively, I guess that it makes sense just by using the axiom of choice.
Start by defining a new sequence $C_ν=\bigcup_{η<ν}B_η$.
This sequence have the same union as the sequence of $B_ν$ and it is increasing in a weak sense, $C_ν=\bigcup_{η<ν}B_η⊆\bigcup_{η<μ}B_η=C_μ$.
You can even make this sequence increasing in the strong sense, let $f:λ→λ$ be a function defined recursively by $f(0)=0$, $f(ξ+1)=\min\{x<λ\mid C_{f(ξ)}\subsetneq C_x\}$ and for limit ordinals $f(β)=\sup_{α<β}f(α)$. Now the sequence $D_ν=C_{f(ν)}$ is strictly increasing, indeed for $ν<μ$ we have $D_ν⊊D_{ν+1}$ by definition, and because the $D$ sequence is a subsequence of the $C$ sequence that is increasing in the weak sense we have $D_{ν+1}⊆D_μ$, so $D_ν⊊D_μ$.