tag:blogger.com,1999:blog-27945613.post373462933076827944..comments2023-06-20T02:06:52.150-06:00Comments on Alethiography: Thomae's Function and IntegrabilityTamhttp://www.blogger.com/profile/18079829842465164437noreply@blogger.comBlogger3125tag:blogger.com,1999:blog-27945613.post-35318159442326506762011-11-05T16:16:57.917-06:002011-11-05T16:16:57.917-06:00Anonymous,
This should be clear by the well-order...Anonymous,<br /><br />This should be clear by the well-ordering principle of positive integers. If 1/q > 1/N, then q < N. Since N is finite, then there only a finite number of possible q such that 0 < q < N. Remember, we are choosing q such that q is an integer, so this works.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-27945613.post-7986307141137462222010-07-05T22:03:27.515-06:002010-07-05T22:03:27.515-06:00Sure.
What I said in my summary above is that, if...Sure.<br /><br />What I said in my summary above is that, if you draw a horizontal line between 0 and 1, there are a finite number of points of the form 1/n above that line, for some natural number n.<br /><br />Let's say you draw the line y = 0.21. Then the 1/n values that work are 1/1, 1/2, 1/3, and 1/4. Any larger n's lead to numbers that are less than 0.21 (for instance, 1/5 is 0.2).<br /><br />Even if you choose a very low value, there are still a finite number. For instance, y = 0.01 is 1/100, so for 1/n to be higher than 0.01, n must be lower than 100. And so on.Tamhttps://www.blogger.com/profile/18079829842465164437noreply@blogger.comtag:blogger.com,1999:blog-27945613.post-63838190663712220632010-07-05T21:34:28.357-06:002010-07-05T21:34:28.357-06:00It's not very easy to realize there's only...It's not very easy to realize there's only a finite number of numbers with f(x)=1/q who are bigger than 1/N<br /><br />its a major step of the proof, so you should say more about itAnonymousnoreply@blogger.com