Proposition 9

Let E be an ordered set, F a subset of E, A a subset of F. If both $\sup_E A$ and $\sup_F A$ exist, we have $\sup_E A \leqslant \sup_F A$. If $\sup_E A$ exists and belongs to F, then $\sup_F A$ exists and is equal to $\sup_E A$.

The first assertion follows from the fact that the set $\mathrm{M}$ of upper bounds of A in F is contained in the set N of upper bounds of A in E, and from Proposition 5. On the other hand, if the least element of N lies in F, then it belongs to M and is clearly the least element of M; this proves the second assertion.