Question

c dns 0c dns 0c uL c 12. Let SR and suppose that s=sup S belongs to S. If ut S, show that sup(SU fu}) sup{s, u}. 13. Show that a nonempty finite set SCR contains its supremum. [Hint: Use Mathematical.

Answer 1

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we will show both side equality observe that u lessthanorequalto sup {s^*, u} & s lessthanorequalto s * lessthanorequalto sup {s *, u} Forall s elementof S thus sup {S Union {u}} lessthanorequalto sup {s *, u} - (i) for other side observe that since s * elementof S thus s * lessthanorequalto sup {S Union {u}} & u lessthanorequalto sup {S Union {u}} since u elementof S Union {u} thus sup {s *, u} lessthanorequalto sup {S Union {u}} - (ii) (i) & (ii) Rightarrow sup {S Union {u}} = sup {s *, u}