February 22nd, 2018, 09:34 AM  #31 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,885 Thanks: 1088 Math Focus: Elementary mathematics and beyond 
Please remain on topic.

February 22nd, 2018, 10:51 AM  #32  
Senior Member Joined: Jun 2015 From: England Posts: 891 Thanks: 269  Quote:
Last edited by skipjack; February 23rd, 2018 at 11:12 AM.  
February 22nd, 2018, 12:45 PM  #33  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,512 Thanks: 2514 Math Focus: Mainly analysis and algebra  Quote:
In the first, the concept of limit is of great significance, because the expression has no value at infinity. We effectively define a value at infinity using the limit. In the second, we don't. The infinite sequence actually exists (in the abstract). The limit is irrelevant to the value at infinity. Last edited by skipjack; February 23rd, 2018 at 11:14 AM.  
February 23rd, 2018, 08:08 AM  #34  
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,603 Thanks: 115  Quote:
Last edited by skipjack; February 23rd, 2018 at 11:13 AM.  
February 23rd, 2018, 08:48 AM  #35 
Global Moderator Joined: Dec 2006 Posts: 19,983 Thanks: 1853  
February 23rd, 2018, 10:25 AM  #36  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,512 Thanks: 2514 Math Focus: Mainly analysis and algebra  Quote:
For example, by your logic, since every finite decimal of the form $0.49999\ldots$ is less than $\frac12$, you must have (using your notation) $$\lim_{n \to \infty}. 499999 = \frac12 \lt \frac12$$ This is obviously nonsense.  
February 23rd, 2018, 10:28 AM  #37  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,512 Thanks: 2514 Math Focus: Mainly analysis and algebra  Quote:
Last edited by skipjack; February 23rd, 2018 at 11:09 AM.  
February 23rd, 2018, 03:45 PM  #38  
Math Team Joined: May 2013 From: The Astral plane Posts: 1,922 Thanks: 774 Math Focus: Wibbly wobbly timeywimey stuff.  Quote:
Dan  
February 24th, 2018, 01:09 AM  #39  
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,885 Thanks: 1088 Math Focus: Elementary mathematics and beyond  Quote:  
February 24th, 2018, 07:41 AM  #40  
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,603 Thanks: 115 
I neglected to mention a crucial number in my previous post: the second number in the abbreviated listing of nplace decimals: For 4place decimals the count should be .0000, .0001,...,.9999. That makes the count easier to visualize. The count (a PROPERTY) of nplace decimals is finite as long as n is finite. As n $\displaystyle \rightarrow$ infinity, the count $\displaystyle \rightarrow$ {countable) infinity. The first number in the list above is .0000....001, FOR ALL N, BY DEFINITION. It has the PROPERTY that it approaches the functional limit zero as n $\displaystyle \rightarrow$ infinity. But that doesn't mean it IS 0. If it meant that it was zero, then the next number in the list would approach (be) zero and the next number would approach (be) 0 etc. The confusion below comes from mixing up the definition of a real number with its properties. By definition of a real number, two real numbers are equal iff they have the same digits FOR ALL n. Period.  Quote:
Quote:
Two real numbers are equal iff they have the same digits for all n. I distinguish between limiting sum of an nplace decimal, which is not unique but rather a property of nplace decimals, and the typographical limit as n \approaches infinity, which is unique. Two real numbers are the same if every digit is the same for all n, not if they approach the same \epsilon \delta limit. Quote:
Function: DEFINITION: Map from elements of one set to another. Continuity: PROPERTY of a function. Limit: PROPERTY of a function. $\displaystyle \epsilon, \delta$ for f(x) or $\displaystyle \epsilon$, M, n>M, for f(n) Of course one can have limits of imaginary numbers. Thanks for your interest in the thread. Last edited by skipjack; February 26th, 2018 at 09:38 AM.  

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