Let $A$ be the set of all functions $f$ with domain $D_f$ a subset of $X$ and range $R_f$ a subset of $Y$ and let $A$ be a partially ordered by extension; that is $g\preceq f$ if and only if $g\subseteq f$. If $\{g_s : s\in S\}$ is a chain in $A$, show that the function $g$ such that $D_g$ equals $\bigcup_{s\in S} D_{g_s}$, and $g(x)=g_s(x)$ for all $x\in D_{g_s}$ is an upper bound for this chain in $A$.
Real Analysis Least upper bound of a set of functions?
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1I edited your "question" using LaTeX. If you click *edit* on it, you can see exactly how this is done. Also note that "questions" which are verbatim copies of textbook/homework problems are considered low-quality material, and are voted on accordingly. See http://meta.math.stackexchange.com/questions/1803/how-to-ask-a-homework-question – 2012-09-09