= Cardinal bound for an increasing chain of countable sets
{title2=$|\bigcup_iX_i|\leq\aleph_1$}
An <increasing chain of countable sets> has union of <cardinality> at most $\aleph_1$. If the union is uncountable, select $\aleph_1$ of its points and one containing member for each. Every countable member misses one selected point, so linear comparison puts it below that point's chosen member. The chosen subfamily is cofinal, and its union has size at most $\aleph_1$ by <infinite cardinal arithmetic>.
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