In Simmons' Introduction to Topology and Modern Analysis, he asks (Problem 8.10) for to show that the cardinal number of the union is less than or equal to that of the index set by making use of Zorn's Lemma. I did this however my proof seems to imply that they have the same cardinality and I wonder if I made an incorrect assumption somewhere:
1- Let F be the family of all classes of countably infinite pair-wise disjoint subsets of I. We impose the partial order relation of class inclusion on F and pick any chain {$C_i$} $\subseteq$ F.
2- A clear upper bound of our chain is $\cup_i C_i$ and we demonstrate that it is in F as follows:
a) It is obvious that it is a class of countably infinite sets.
b) Suppose $c_1\in C_1$ and $c_2\in C_2$. Then since $C_1$ and $C_2$ are in {$C_i$}, it follows that either $C_1 \subseteq C_2 $ or $C_2 \subseteq C_1 $. WLOG let $C_1 \subseteq C_2 $. Then $c_1 \in C_2$ and $c_2 \in C_2 \implies c_1 \cap c_2 = \emptyset$. Thus $\cup_i C_i \in$ F
3- We apply Zorn's Lemma and construct the maximal class $M$={$m_j$} $\in$ F and $j \in J$, the index set of $M$. Clearly, $\cup_j m_j \subseteq I $.
a) If $\cup_j m_j \subset $ I then I$-\cup_j m_j$ is finite else $M$ is not maximal. We can therefore establish a one to one correspondence between $\cup_j m_j$ and I. (Sketch: construct a set $n$ = $m_k$ $\cup$ (I$-\cup_j m_j$) which should not change its cardinality and take the union of $n$ with the other of the sets in $M$ and thus establish a one-to-one correspondence with $I$)
b) Now construct $\chi_j$ = {$\cup_k$$X_k$ : $k$ $\in$ $M_j$}. Since the $M_j$ are countably infinite and the $X_i$ are countable, it follows that the $\chi_j$ are countably infinite and thus one may establish a one-to-one correspondence between the $\chi_j$ and the $M_j$ and by extension between $\cup_j \chi_j$ and $\cup_jM_j$ and so between $\cup_j \chi_j$ and I. But $\cup_j \chi_j$ = $\cup_iX_i$.