How to prove closure of $\mathbb{Q}$ is $\mathbb{R}$

Presumably, you mean the closure of $\mathbb{Q}$ in $\mathbb{R}$ under the usual topology.

It suffices to show that for every real number $r$ and every $\epsilon\gt 0$, there is at least one rational $q$ which is "$\epsilon$-close" to $r$ (that is, $|r-q|\leq \epsilon$), since this will show that every open ball around $r$ contains a rational. This shows that the complement of $\mathbb{Q}$ has empty interior, so the closure of $\mathbb{Q}$ is all of $\mathbb{R}$.

To prove that, let $r$ be a real number, and let $\epsilon\gt 0$. Then there exists a natural number $n$ such that $\frac{1}{10^n}\lt \epsilon$. Write $r$ in decimal expansion, and let $q$ be the rational that has the same decimal expansion as $r$ up to the $n$th digit after the decimal point, and terminates there. This is a rational number, since its decimal expansion terminates.

Now note that $|r-q|\leq \frac{1}{10^n} \lt \epsilon$, proving that there is a rational that is $\epsilon$-close to $r$.

This question really does not make a whole lot of sense. In topology closures only makes sense in some ambient space. Some examples will probably make this clear.

• $\mathbb{Q}$'s closure in $\mathbb{Q}$ is $\mathbb{Q}$ itself.
• $\mathbb{Q}$'s closure in $\mathbb{R}$ is $\mathbb{R}$.
• $\mathbb{Q}$'s closure in $\mathbb{Q}(\sqrt{2})$ is $\mathbb{Q}(\sqrt{2})$.

And this doesn't even go into embeddings of $\mathbb{Q}$ in e.g. the unit circle. What you probably meant to ask was how to prove that the completion of $\mathbb{Q}$ is $\mathbb{R}$. For that you should take a look at this question.

In some sense, the denseness of $\Bbb Q$ in $\Bbb R$ is implicit in the very same construction of $\Bbb R$.

There are a few equivalent ways to construct $\Bbb R$. One of them is to endow $\Bbb Q$ with a structure of metric space by defining the distance function $d(a,b)=|a-b|$ and define $\Bbb R$ to be the completion of the metric space $({\Bbb Q},d)$.

This is but an instance of a general construction: given any matric space $(X,d)$ one can construct its completion $\tilde X$ as the set of certain equivalence classes of Cauchy sequences in $X$. It turns out that one can endow $\tilde X$ with a natural metric space stucture such that: (1) $X$ embeds isometrically into $\tilde X$ and the image is a dense set, (2) $\tilde X$ is complete, and (3) if $\tilde Y$ is some metric space satisfying the just mentioned properties, then there is a canonical isometry of metric spaces between $\tilde X$ and $\tilde Y$ which is the identity on $X$.

Incidentally, it is possible to define distances in $\Bbb Q$ which are not equivalent to the one given by the usual absolute value (as above) but depend on the choice of a prime number $p$ and are defined arithmetically. The corresponding completions, the so-called fields of $p$-adic numbers ${\Bbb Q}_p$, are metric spaces not equivalent to $\Bbb R$ in which $\Bbb Q$ sits densely. A famous theorem by Ostrowski says that $\Bbb R$ and the ${\Bbb Q}_p$ are, up to equivalence, the only completions of $\Bbb Q$.