推 wengying :謝謝k大^^ 02/27 07:59
※ 引述《wengying (ammy)》之銘言：
: Let {x_n be a convergent sequence in a metric space and lim(x_n)=x.
: n->無窮
: Is the set A={x_1,x_2,x_3,...}聯集{x} is cpmpact
Let {V_α} be an arbitrary open covering of A.Then there exists an open set V
in {V_α} s.t. x屬於V.
Since V is open,we can choose ε>0 s.t. B(x,ε)ㄈV.
Since x_n→x as n→∞,there exists a positive integer N s.t. x_n屬於B(x,ε)
for all n>N.So,{x,x_(N+1),x_(N+2),...}ㄈV.
For each term of the remained terms {x_1,x_2,...,x_N},we are able to find
at most N open sets,say V_1,V_2,...,V_N,in {V_α} s.t.
{x_1,x_2,...,x_N}ㄈV_1∪V2∪…∪V_N.
Thus,{V,V_1,V_2,...,V_N} is a finite subovering of {V_α} which is what we
want.
i.e.,A is compact.
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律:知道嗎?聽說我們的歌被海外的電視台所錄用耶!看來我們離武道館不遠了
唯:真的嗎?那真的是太好了，我一直夢想能在武道館彈著吉太，好高興
釉:小唯能高興真的是太好了，呵呵~
澪:拜託!那個明明是盜用不是錄用，你們怎麼還這麼高興?
律、唯、釉:啊?什麼? 輕音部
澪:絕望啦!我對盜用錄用分不清楚的輕音部社員們絕望啦! 邁向武道館之路
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