[208](Solved on 2022-11-24) Let therefore \(Aββ\) be not empty, let \(l β β βͺ \{ +β\} \); you can easily demonstrate the following properties:
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\(\sup A β€ l\) |
\(β xβ A,xβ€ l\) |
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\(\sup A {\gt} l\) |
\(β xβ A,x{\gt} l\) |
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\(\sup A {\lt} l\) |
\(β h{\lt}l , β xβ A,xβ€ h\) |
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\(\sup A β₯ l\) |
\(β h{\lt}l, β xβ A,x{\gt} h\) |
the first and third derive from the definition of supremum,
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the second and fourth by negation; in the third we can conclude equivalently that \(x{\lt}h\), and in the fourth that \(xβ₯ h\).
If \(lβ +β\) then also we can also write (replacing \(h=l-\varepsilon \))
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\(\sup A {\lt} l\) |
\(β \varepsilon {\gt}0 , β xβ A,xβ€ l-\varepsilon \) |
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\(\sup A β₯ l\) |
\(β \varepsilon {\gt}0, β xβ A,x{\gt} l-\varepsilon \) |