Anonymous
Anonymous asked in 科學數學 · 2 decades ago

一個高等微積分的證明題....請各位大大幫個忙

Assume that S(R is a bounded set.

Prove that sup S and inf S both adhere to S

請各位大大幫忙了!!

1 Answer

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  • 2 decades ago
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    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

    PS:不會打數學符號,自己改一下吧...有不懂在問

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