Anonymous

# 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

Rating

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