推 suhorng :某則 D+(1/n) 就是個 lower bound, 違背 greatest 12/19 20:45
→ suhorng :否則 12/19 20:45
謝謝大大~~!!!
※ 編輯: pop10353 來自: 124.9.198.11 (12/19 21:10)
S in R
1). nonempty
2). bound below
3). closed
Q: Show that inf(S) 屬於 S
<pf>
Since (1),(2), by greast lower bound property,
there is an D=inf(S) in R.
We know D+(1/n), for all n=1,2,3...
is not lower bounds.
*So there must be points Xn in S with D≦Xn﹤D+(1/n)
So |Xn-D|< 1/n.
1/n ->0 ,we conclude Xn-> D=inf(S).
Since <Xn> is a sequence in S converging to inf(S),
(By x屬於cl(S) iff there is a sequence Xn in S with Xn->x)
We have D=inf(S)屬於cl(S)=S
(cl(S)=S since S is closed.
想問*那，為什麼在S中一定會有個那些點在那個範圍內 ？
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