# 高等微積分(緊致性和完備性)

Rating
• L
Lv 7

Claim :

Every sequence in T has a convergent subsequence with limit in T.

pf of claim :

Let {x_n} be a sequence in T which has no convergent subsequence, then for each x in T, there exist B(x,r_x) so that B(x,r_x) contains finitely many elements of {x_n} (otherwise, x will be a limit of some subsequence of {x_n}). The union of all B(x,r_x) convers T, since T is compact, there are finitely many B(x,r_x) so that their union convers T, this implies {x_n} is finite, contradiction. The claim is true.

Since a Cauchy sequence which has a convergent subsequence must converge and the limit as same as the limt of its subsequence.

From above, the result follows.

我本來想出你說的第二題,但比較難寫而作