I have a simple question that I haven't been able to prove but I think is true, I hope you can help me. Suppose I have the POSET $(B,\subseteq)$ where each element of B has cardinality of $2^{\aleph_0}$. If $C$ is a chain of elements of $B$, can I conclude that $\bigcup C$ has cardinality $2^{\aleph_0}$ also?
Thank you
Nope; this is false with any infinite cardinal in place of $2^{\aleph_0}$. Here's a quick and easy proof using Zorn's lemma. Let $\kappa$ be an infinite cardinal and let $X$ be any set of cardinality greater than $\kappa$. Let $B$ be the set of all subsets of $X$ of cardinality $\kappa$. By Zorn's lemma, the poset $B$ contains a maximal chain $C$. Now suppose $\bigcup C$ has cardinality $\kappa$. Let $x\in X\setminus\bigcup C$ and let $b=\{x\}\cup\bigcup C$. Then $b$ strictly contains every element of $C$ and has cardinality $\kappa$, so $C\cup\{b\}$ is a chain in $B$ strictly containing $C$. This violates maximality of $C$. Therefore $\bigcup C$ must have cardinality greater than $\kappa$.
You can say something much more precise using the theory of well-orderings. For any infinite cardinal $\kappa$, note that the cardinal $\kappa^+$ (the least cardinal greater than $\kappa$) is a union of a chain of sets of size $\kappa$, namely the chain consisting of all ordinals $\alpha$ such that $\kappa\leq\alpha<\kappa^+$. Conversely, if $C$ is a chain of sets of size $\kappa$, let $W\subseteq C$ be a well-ordered cofinal subset of $C$. Each proper initial segment of $W$ must have cardinality at most $\kappa$, since for any ordinal $\alpha$, the $\alpha$th element of $W$ has at least $|\alpha|$ elements (since for each successor $\alpha$ you must add at least one new element that was not in any earlier element of $W$). It follows that the well-ordered set $W$ has order type at most $\kappa^+$, and in particular $W$ has cardinality at most $\kappa^+$. Thus $\bigcup C=\bigcup W$ has at most $\kappa^+\cdot\kappa=\kappa^+$ elements.
In conclusion, if $C$ is a chain of sets of cardinality $\kappa$, the best upper bound you can get on the cardinality of $\bigcup C$ is that it is at most $\kappa^+$.