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Unformatted text preview: 1.104 The following sequence approximates the square root of any positive number 1 = +1 = 1 2 ( + ) 1.105 Let . If , then is the limit of the sequence ( ,,,... ). If / , then is a boundary point of . For every , the ball ( , 1 / ) contains a point . From the sequence of open balls ( , 1 / ) for = 1 , 2 , 3 ,... , we can generate of a sequence of points which converges to . Conversely, assume that is the limit of a sequence ( ) of points in . Either and therefore . Or / . Since , every neighborhood of contains points of the sequence. Hence, is a boundary point of and . 1.106 is closed if and only if = . The result follows from Exercise 1.105....
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- Fall '10