b****t 发帖数: 114 | 1 A closed set A \in R^n and a compact set B \in R^n
show that A + B is closed, where A + B = {z: z=x+y, x \in A, y \in B, and z
\in R^n}.
provide a example that if B is closed but not compact set, then the result
does not hold.
谢谢!
beet | B****n 发帖数: 11290 | 2 1. Assume z_n\in A+B and z_n->z,
there exists x_n\in A, y_n\in B such that z_n=x_n+y_n.
Since B is compact, there exists a subsequence of y_n' such that y_n'->y\in
B
so there exists a subsequence x_n'->x\in A
so z=x+y\in A+B
2. choose A={1,2,3,...} B={-1,-2+1/2.-3+1/3,...}
z
【在 b****t 的大作中提到】 : A closed set A \in R^n and a compact set B \in R^n : show that A + B is closed, where A + B = {z: z=x+y, x \in A, y \in B, and z : \in R^n}. : provide a example that if B is closed but not compact set, then the result : does not hold. : 谢谢! : beet
| u*****n 发帖数: 28 | 3
in
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~这是为啥?
~~~~~~~~~~~~~~~~这个是怎么得出来的?
【在 B****n 的大作中提到】 : 1. Assume z_n\in A+B and z_n->z, : there exists x_n\in A, y_n\in B such that z_n=x_n+y_n. : Since B is compact, there exists a subsequence of y_n' such that y_n'->y\in : B : so there exists a subsequence x_n'->x\in A : so z=x+y\in A+B : 2. choose A={1,2,3,...} B={-1,-2+1/2.-3+1/3,...} : : z
| B****n 发帖数: 11290 | 4 x_n'=z_n'-y_n'->z-y. Because A is closed z-y is in A
Define x=z-y, so z=x+y is in A+B
【在 u*****n 的大作中提到】 : : in : ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~这是为啥? : ~~~~~~~~~~~~~~~~这个是怎么得出来的?
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