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 October 12th, 2019, 12:23 PM #1 Senior Member   Joined: Dec 2015 From: Earth Posts: 823 Thanks: 113 Math Focus: Elementary Math Real numbers in (0,1) Show whether the product of all real numbers in interval (0,1) converges or diverges ? October 12th, 2019, 01:52 PM #2 Global Moderator   Joined: May 2007 Posts: 6,852 Thanks: 743 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. Thanks from greg1313, topsquark, tahirimanov19 and 1 others Last edited by skipjack; October 12th, 2019 at 02:13 PM. Reason: typo October 14th, 2019, 06:19 PM #3 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,696 Thanks: 2681 Math Focus: Mainly analysis and algebra 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). October 14th, 2019, 07:54 PM   #4
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Quote:
 Originally Posted by v8archie 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. October 15th, 2019, 12:58 AM   #5
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Quote:
 Originally Posted by v8archie 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. Tags numbers, real Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post zylo Topology 14 May 10th, 2017 02:57 AM Elektron Math 4 May 7th, 2017 12:47 PM Congeniality Math Books 2 June 10th, 2015 09:25 AM thehurtlooker Algebra 3 April 9th, 2013 01:58 AM greg1313 Applied Math 2 August 11th, 2011 04:45 AM

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