Say we have the set $[1,5]$, this being a closed set, we know the infimum of supremum of this set is inside the set. Shouldn't the supremum in this case be $5$ and the infimum be $1$? Also in general, in a closed set [a,b], can we say supremum and infimum be $b$ and $a$ respectively?
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Yes. In short, by definition $x\leq b$ for all $x\in [a,b]$, thus $b$ is an upper bound. Also, if $S$ is any upper bound of $[a,b]$ then in particular $b\leq S$, as $b$ is itself an element of $[a,b]$. So $b$ is the smallest upper bound. – Mark Jul 18 '23 at 18:01
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@Mark Thanks! Kust another question, say instead of closed interval, we have ($a,b$) , the can we say, the supremum and infimum is $a- \epsilon, b+ \epsilon$ for some $\epsilon>0$ which can be arbitrarily small – Ellie_Wong Jul 18 '23 at 18:09
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No, supremum and infimum are unique. In this case it can be proved that they are still $b$ and $a$ respectively, though this time they don't belong to the set. – Mark Jul 18 '23 at 18:29