In Reinhard Diestel's book "Graph Theory" (5th ed.) there is a chapter on infinite graphs (chapter 8).
In that chapter Diestel states the following fact related to Menger's Theorem:
Proposition 8.4.1 Let $G$ be any graph, $k \in \mathbb{N}$, and let $A, B$ be two sets of vertices in $G$ that can be separated by $k$ but no fewer than $k$ vertices. Then $G$ contains $k$ disjoint $A-B$ paths.
Directly below the proof of that proposition, he states:
When k is infinite, however, the result suddenly becomes trivial.
I interpret this as the following reformulation of the Proposition:
If two vertex sets $A$, $B$ of an (infinite) graph $G$ can not be separated by finitely many vertices, then $G$ contains infinitely many disjoint $A$-$B$ paths.
He then argues why that is:
Let $\mathcal{P}$ be any maximal set of disjoint $A-B$ paths in $G$. Then the union of all these paths separates $A$ from $B$, so $\mathcal{P}$ must be infinite. But then the cardinality of this union is no bigger than $|\mathcal{P}|$. Thus, $\mathcal{P}$ contains $|\mathcal{P}| = | \bigcup \mathcal{P} | \geq k$ disjoint $A-B$ paths, as desired.
I do not understand why we can immediately conclude that $\mathcal{P}$ is infinite in the second sentence. What if the number of disjoint $A-B$ paths is finite, but some of the paths are infinite? Then $\bigcup \mathcal{P}$ still contains an infinite number of vertices, including those which separate $A$ and $B$.
For example, there could be a non-disjoint set $\mathcal{Q}$ of $A-B$ paths that have an infinite number of distinct intersection vertices with some infinite paths of $\mathcal{P}$ while still only intersecting with finitely many paths of $\mathcal{P}$ in total.
I have been thinking about this for quite a long time and I suppose I am just missing some really obvious fact here. Maybe I should also point out that I have very little experience with infinite graphs and am still learning the basics of graph theory.
Maybe you can point me in the right direction?