Small Understanding of [0,1]/\Q in R

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Discussion Overview

The discussion revolves around the concept of the set [0,1] ∩ Q (the rational numbers in the interval [0,1]) and its closure properties in the real numbers R. Participants explore sequences of rational numbers and their convergence to irrational numbers, aiming to understand why [0,1] ∩ Q is not closed in R.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification, Debate/contested

Main Points Raised

  • One participant suggests constructing sequences of rational numbers of the form m/n to explore their limits within [0,1] ∩ Q.
  • Another participant proposes using a sequence derived from an irrational number's decimal expansion to illustrate convergence to an irrational limit.
  • A specific example involving the irrational number sqrt(2)/2 is provided, demonstrating how a sequence of rational approximations converges to this irrational number.
  • One participant advises focusing on finding rational sequences that converge to an irrational number rather than attempting to create a sequence of rationals that might converge to an irrational limit.
  • There is a correction regarding the identification of the irrational number, clarifying that it is sqrt(2)/2, not sqrt(22).
  • A participant expresses gratitude for the clarification and acknowledges the need to show that perturbing a rational number leads to irrational numbers, referencing the sqrt(2)/2 example.
  • Another participant questions the sufficiency of the argument presented in the discussion.

Areas of Agreement / Disagreement

Participants generally engage in a constructive dialogue, but there are differing approaches to understanding the closure of [0,1] ∩ Q and the methods for demonstrating convergence to irrational numbers. No consensus is reached on the best approach or the sufficiency of the arguments presented.

Contextual Notes

Some participants express uncertainty about the correct sequences to use and the implications of their convergence. There are also unresolved questions regarding the clarity and sufficiency of the arguments made in the discussion.

Sumanta
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Hi,

This is not a homework question since I am out of college for a long time.

I was trying to understand the following that [0,1] /\ Q is not closed in R.
My understanding is that u must take a sequence (since this is a metric space) of the form m/n s.t m < n and create a sequence.

So I was trying to construct sequences like 1/2, 2/3, 3/4 but they seemed to be all ending within [0,1] /\ Q. I am not sure but do I have to take a sum or sth but I am not sure how to prove it.
 
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(I am assuming that Q means rational nos. and R means real nos.). Take any irrational number between 0 and 1 and let the rational number sequence be the the sequence where the nth term is the truncation of the decimal expansion of the irrational after n decimal places.
 
Just giving an example of what mathman said: [itex]\sqrt{2}{2}[/itex] is irrational and is 0.70710678118654752440084436210485...
Each number in the sequence 0.7, 0.70, 0.707, 0.7071, 0.70710, 0.707106,... is a rational number because it is a terminating decimal; but the sequence as a whole converges to the irrational number [itex]\frac{\sqrt{2}}{{2}}[/itex]
 
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You're working from the wrong direction. Rather than trying to come up with a sequence of rationals and hope that its limit is irrational... you should pick the irrational, and then try and find a sequence of rationals that converges to it.
 
Thanks a lot for clearing the lacunae in my understanding. I think the basic idea is that u need to show that if u wiggle the rational number a bit u will fall into the set of irrational numbers, which is clearly proved by the sqrt(22) example.

Thank u a lot for the same.
 
Actually that was sqrt(2)/2 (how could the square root of 22 be less than 1?). You should read what Hurkyl said. The simplest irrational you can think of in the interval is sqrt(2)/2, and constructing a sequence that converges to it is easy, as shown by HallsofIvy.
 
'Cause I'm nothing if not simple!
 
Do you understand why this argument suffices?
 

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