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Math Help - Density of Q in R

  1. #1
    Senior Member Danneedshelp's Avatar
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    Density of Q in R

    How do I prove the following claim?

    "Given any y\in\mathbb{R}, there exists a sequence of rational numbers that converges to y"

    I'd assume, I would need to use the fact that Q is dense in R (not the topological version I'm trying to prove) and that, given any epsilon neighborhood around some real number x, I can find a rational number in the interval. I just need help formalizing these thoughts and constructing a solid proof.

    Thanks
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  2. #2
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    Quote Originally Posted by Danneedshelp View Post
    How do I prove the following claim?

    "Given any y\in\mathbb{R}, there exists a sequence of rational numbers that converges to y"

    I'd assume, I would need to use the fact that Q is dense in R (not the topological version I'm trying to prove) and that, given any epsilon neighborhood around some real number x, I can find a rational number in the interval. I just need help formalizing these thoughts and constructing a solid proof.

    Thanks

    If you "would need the fact that \mathbb{Q} is dense in \mathbb{R}" then there's already nothing to prove.

    You can try the following: if r\in\mathbb{R} is rational there's nothing to prove, otherwise we can write r=a_0.a_1a_2... , an infinite non-periodic decimal expression, so you can now define x_0=a_0\,,\,\,x_1=a_0.a_1\,,\,\,...x_n=a_0.a_1...a  _n,.... It's now simple to check that \lim_{n\rightarrow\infty}x_n=r.

    The above is somewhat unsatisfying though, since we have first to know the decimal expression of the irrational in order to be able to define the rational sequence that converges to it.

    Tonio
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