Help on formalization of the proof: If $(a,b] = \bigcup_{n=1}^{\infty}(a_n,b_n]$ (disjoint union), then we can supose $a_{n+1}=b_n \forall n \geq 1$

If $a<b$ are extended numbers for which $(a,b] = \bigcup_{n=1}^{\infty}(a_n,b_n]$, where $((a_n,b_n])_{n \geq 1}$ is a sequence of disjoint intervals such that $a_n \leq b_n$, so it is clear that:

intuitively speaking, “the edges of the intervals must touch (i.e. be equal)”, otherwise their union would have to have a “hole” and could not be an interval,

In other words, we can find a permutation $j: \mathbb{N} \rightarrow \mathbb{N}: n \mapsto j_n$ for which $a_{j_n+1}= b_{j_n} \forall n\geq 1 $

Even though it seems obvious, this seems to be part of the group of obvious things that are difficult to prove (at least I found it difficult to formalize the proof). Could someone help me to formalize the argumentation?