Anonymous asked in 科學數學 · 2 decades ago


Assume that S(R is a bounded set.

Prove that sup S and inf S both adhere to S


1 Answer

  • 2 decades ago
    Favorite Answer

    let inf S = h

    Suppose h is not adhereent point of S

    --> exist a d>0 such that (h-d,h+d)交集S = 空集合

    --> s<=h-d or h+d<=s

    h is a lower bound of S --> h<s for all s屬於S

    -->h+d<=s for all s屬於S

    let k = h+d/2

    --> k<s for all s屬於S

    -->k is a lower bound of S

    and h<k (矛盾) (因為h = inf S =max{h| h is a lower bound of S})

    --> h is a adherent point of S


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