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Math Help - Density of Q in R

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    Density of Q in R

    Can someone prove (and explain step by step the proof) the density of Rationals in Real numbers?

    By the way, in my notebook they gave the density of rationals in real numbers as definition, my second question is, definitions are things that we define and we can't prove them so why is there a proof? Maybe it's not a definition but a statement ? Or do definitions have proofs too? Thank you in advance.
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    Re: Density of Q in R

    Quote Originally Posted by davidciprut View Post
    Can someone prove (and explain step by step the proof) the density of Rationals in Real numbers?
    I will not do that. But I will give you an outline.
    1) If r\in\mathbb{R} then (\exist j\in\mathbb{Z})[j\le r<j+1]. (Think floor function).

    2) If y-x>1 then (\exists K\in\mathbb{Z})[x < K< y],
    Use 1) note that \left\lfloor x \right\rfloor  \le x < \left\lfloor {x } \right\rfloor +1 or \left\lfloor x \right\rfloor +1 \le x+1 < y So  x < \left\lfloor {x } \right\rfloor+1<y .

    3) If 0<t<s then \left( {\exists J \in\mathbb{ Z}^+} \right)\left[ {J > \frac{1}{{s - t}}} \right] of course that means Js-Jt>1. Use #2.

    4) You get Js>K>Jt. Divide by J.
    Last edited by Plato; November 15th 2013 at 03:01 PM.
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    Re: Density of Q in R

    I am puzzled as to what you are trying to say.

    Can someone prove (and explain step by step the proof) the density of Rationals in Real numbers?
    Prove what about density of the Rationals in the Real numbers? "The density of the Rationals in the Real numbers" is a noun phrase. We prove statements, not phrases.

    By the way, in my notebook they gave the density of rationals in real numbers as definition, my second question is, definitions are things that we define and we can't prove them so why is there a proof? Maybe it's not a definition but a statement ? Or do definitions have proofs too? Thank you in advance.
    We define nouns, we prove statements. We might include some properties in the definition of a noun, rather than proving them. But, exactly what was this definition you are referring to? I know a definition of one set being "dense" in a superset, given some topology. Give such a definition, one would then have to prove that this definition applied to the rational numbers as a subset of the real numbers, given the usual topology.

    For example, we can say that any set, X, is "dense in the real numbers" if and only if, given any real number, y, and real number [itex]\lambda> 0[/tex], there exist a number, x, in X, such that [itex]|x- y|< \lambda[/itex]. We could then show, using that definition, that Q is dense in R.
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