Proposition 6

Let $(x_\iota)_{\iota \in I}$, $(y_\iota)_{\iota \in I}$ be two families of elements of an ordered set $\mathrm{E}$, indexed by the same set $\mathrm{I}$, and such that $x_\iota \leqslant y_\iota$ for all $\iota \in \mathrm{I}$. If both families have a least upper bound in $\mathrm{E}$, then $\sup\limits_{\iota \in I} x_\iota \leqslant \sup\limits_{\iota \in I} y_\iota$.

For $a = \sup\limits_{\iota \in I} y_\iota$ is an upper bound of the set of the $y_\iota$, so that $x_\iota \leqslant y_\iota \leqslant a$ for all $\iota \in \mathrm{I}$, and hence $\sup\limits_{\iota \in I} x_\iota \leqslant a$.