Proving Existence of Sequence Converging to x in R

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Discussion Overview

The discussion revolves around the existence of a sequence {a_k} that converges to a real number x, specifically focusing on sequences within the rational numbers Q. Participants explore different approaches to construct such a sequence and the implications of the density of Q in R.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant suggests using the definition of the limit of a sequence to find a sequence that converges to an arbitrary x in R.
  • Another participant proposes the constant sequence a_k = x as a straightforward example of a sequence converging to x.
  • There is a clarification that the sequence must be in Q, prompting further discussion on the implications of this restriction.
  • Some participants discuss the density of Q in R, questioning whether this property can be used to find a sequence converging to x.
  • One participant raises the idea that for any x in R, a rational number can be found in the interval (x-1/n, x+1/n), which could help in constructing a rational sequence.
  • Another participant suggests using successive truncations of x's decimal expansion as a method to create a sequence of rationals that converges to x.

Areas of Agreement / Disagreement

Participants express various approaches to constructing a converging sequence, but there is no consensus on a single method. Multiple competing views and techniques are presented, indicating an unresolved discussion.

Contextual Notes

The discussion includes assumptions about the properties of rational numbers and their relationship to real numbers, particularly regarding density and convergence. There are also unresolved questions about the specific requirements for the sequence.

Who May Find This Useful

This discussion may be of interest to those studying real analysis, particularly in understanding sequences, limits, and the properties of rational numbers in relation to real numbers.

autre
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I asked this question before but I totally misunderstood what it was asking. Basically, I need to find that there exists a sequence {a_k} such that it converges to x for some x in R.

Since the real numbers are equivalence classes of convergent Cauchy sequences the result seems fairly obvious, but I was thinking of instead of dealing with equivalence classes I could use the definition of the limit of a sequence and have x be the limit of {a_k}.

I'm having some trouble starting up, though. Do I have to find a sequence that must converge to some arbitrary x to prove its existence?

Thanks!
 
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Take the constant sequence [itex]a_k=x[/itex]. Surely, the sequence

[tex]x,~x,~x~,...[/tex]

converges to x...
 
Sorry, forgot to mention that the sequence is in Q.
 
autre said:
Sorry, forgot to mention that the sequence is in Q.

Well, what can you go from?? Can you use that Q is dense in R?? Or must you use the definition of R?
 
does the fact that Q is dense in R tell me that if I take any member of {s_n} it can exist in an interval of, say, (x-1/n,x+1/n) ?
 
autre said:
does the fact that Q is dense in R tell me that if I take any member of {s_n} it can exist in an interval of, say, (x-1/n,x+1/n) ?

Q being dense says that for each x in R, there exists a rational in (x-1/n,x+1/n). You can use this to construct a rational sequence.
 
autre said:
Sorry, forgot to mention that the sequence is in Q.

Can you just take successive truncations of x's decimal expansion? For example

3, 3.1, 3.14, 3.141, 3.1415, ... is a sequence of rationals converging to pi.
 

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