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October 12th, 2019, 12:23 PM   #1
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Real numbers in (0,1)

Show whether the product of all real numbers in interval (0,1) converges or diverges ?
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October 12th, 2019, 01:52 PM   #2
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Trivial: Consider a subset consisting of all the reciprocals of the integers. The product is $\displaystyle\lim_{n\to \infty} \frac{1}{n!} =0$. All other contributions to the product are $\le 1$, making it go to zero faster.

Last edited by skipjack; October 12th, 2019 at 02:13 PM. Reason: typo
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October 14th, 2019, 06:19 PM   #3
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Here's a problem though. How do you form the product of uncountably many numbers? A product is formed by multiplying numbers, but that operation is only defined as a binary operation. We can chain them together, but that at best only gives us countably many (as in Mathman's subset).
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October 14th, 2019, 07:54 PM   #4
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Quote:
Originally Posted by v8archie View Post
Here's a problem though. How do you form the product of uncountably many numbers?
That may be a problem in general, but it's not going to affect the outcome of the question at hand. Once you've got down to zero with a countably infinite set of values, the rest aren't going to make it non-zero, whether you can figure out how to multiply them in or not.
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October 15th, 2019, 12:58 AM   #5
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Quote:
Originally Posted by v8archie View Post
Here's a problem though. How do you form the product of uncountably many numbers? A product is formed by multiplying numbers, but that operation is only defined as a binary operation. We can chain them together, but that at best only gives us countably many (as in Mathman's subset).
This is a crucial issue of course. Otherwise the entire problem is ill defined. The only definition of an uncountable product I know is in terms of nets. We would say that
$$\prod_{x\in (0,1)} x = L$$
if and only if for each $\varepsilon>0$, there is some finite subset $F_0\subseteq (0,1)$ such that if $F$ is any finite subset of $(0,1)$ with $F_0\subseteq F$, then
$$\left| \prod_{x\in F} x - L\right|<\varepsilon.$$
With this definition, you can indeed prove the product is $0$.

Obviously other definitions are possible, but this is a really common one.
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