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I'm just an interested laymn, and I'm trying to improve my knowledge in some areas where I'm weak. To this end, I found that Shilov's Elementary Real and Complex Analysis was highly recommended, and the Dover edition was available for only ten bucks, so how could I go wrong? But it didn't take me long to get stuck.

On page 18, corollary 1.74, he says that in considering an infinite series of half open intervals of real numbers, each a half sized subset of the previous, the overall intersection can be empty.

[tex](0,y] \supset (0,\frac{y}{2}] \supset ... \supset (0,\frac{y}{n}] \supset ... [/tex]

This makes no sense to me. A set can only be a subset of another if it contains elements in common, so the intersection of two such sets cannot be empty. Since the reals are infinitely dense, even the smallest interval, no matter how many times you cut it in half, contains an infinite number of points, right? Isn't the intersection the entire contents of the latest subset, wich is never enpty? Plus, he gets this from Theorem 1.73: given arbitrary real numbers x>0 and y>0, there exists an integer n>0 such that y/n < x. I understand that, but don't see at all how it asserts the corollary. Help! I'm hesitant to go on without resolving this, lest I learn a misunderstandings or reach false conclusions. Does anyone here have this book? Can anyone clear up my confusion?

Note that this isn't homework, and its not for any class. I'm just curious. Thanks!

Fixed LaTex: direction of subset symbols

On page 18, corollary 1.74, he says that in considering an infinite series of half open intervals of real numbers, each a half sized subset of the previous, the overall intersection can be empty.

[tex](0,y] \supset (0,\frac{y}{2}] \supset ... \supset (0,\frac{y}{n}] \supset ... [/tex]

This makes no sense to me. A set can only be a subset of another if it contains elements in common, so the intersection of two such sets cannot be empty. Since the reals are infinitely dense, even the smallest interval, no matter how many times you cut it in half, contains an infinite number of points, right? Isn't the intersection the entire contents of the latest subset, wich is never enpty? Plus, he gets this from Theorem 1.73: given arbitrary real numbers x>0 and y>0, there exists an integer n>0 such that y/n < x. I understand that, but don't see at all how it asserts the corollary. Help! I'm hesitant to go on without resolving this, lest I learn a misunderstandings or reach false conclusions. Does anyone here have this book? Can anyone clear up my confusion?

Note that this isn't homework, and its not for any class. I'm just curious. Thanks!

Fixed LaTex: direction of subset symbols

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