Compactness; infinmum

kingwinner

我唔係好明佢個 hint 講乜...
有人可以教下我ma?

唔該哂

Naozumi

即係話,先搵兩個 sequence xn, yn令到 d(xn,yn) -> d(X,Y)
然後證明 xn有convergent subsequence xnk
但係 xnk converage唔代表 ynk converge
所以最後要搵個 'sub-subsequence' xnkm令到 ynkm 都converge

kingwinner

但問題係點搵 xn, yn, 同埋個 convergent subsequence xn_k??

[ 本帖最後由 kingwinner 於 10-2-23 04:52 PM 編輯 ]

Naozumi

[ 本帖最後由 Naozumi 於 10-2-23 10:18 PM 編輯 ]

kingwinner

Thx, 你 d 數真係好勁

Part b 有 counterexample 嗎? 點 prove 個 infinmum is not achieved?

Q1b) Give an example to show that this can fail if neither of X nor Y is bounded.

Naozumi

太過獎了,你再問深入d,我都唔識.
counter-example 要度下,未想到住

kingwinner

b) Let X, Y be subsets of R^2.
X={(n,1/n): n E N}
Y={(n,-1/n): n E N}

呢個得ma?
但我唔識點 prove infinmum is not achieved...

Naozumi

都唔係closed sets

kingwinner

點解唔係 closed ??

咁 X={(x,y): y=0}, Y={(x,y): y=1/x} 得唔得?