Anonymous

# 高等微積分!!

Q: Let O be an open subset of R^n that contains the closed bounded set K. Show that there is a positive number s such that if J is a generalized rectangle that has diameter less than s and contains a point of K,then J is contained in O.(hint:Argue by contradiction.)

Rating

Proof: Suppose otherwise,then for each positive integer n, such that J is a generalized rectangle that has diameter<1/n,and K∩J≠空集合

但是存在 x_n in J but x_n does not in O

Let y€K∩J,J is compact, so the sequence <x_n> has a convergent subsequence,say z_n->z in J

y€J, so we have ||y-z||<1/n

On the other hand,y€O, there is ε>0 such that B(yε) is contained in O

Choose N€|N so that 1/N<ε, then z€O

Hence for n large enough, we get z_n is belongs to O

2009-06-09 21:04:57 補充：

有一行應該是B(y,ε) is contained in O

2009-06-09 21:07:23 補充：

這題也可以直接

2009-06-09 21:07:45 補充：

證明