Euclidean distance on R and Q I have to answer the following three questions.
Given $\epsilon > 0$, a metric space $(V,d)$ and a element $a \in V$
i) Show that for all $x \in B(a;\epsilon)$ there exists a $n \in \mathrm{N}$ s.t. $B(x; \frac{1}{n}) \subset B(a; \epsilon)$
ii) Show that when $V=\mathbb{R}^p$, with euclidean distance, there exists a point $q \in \mathbb{Q}^p$ with $a \in B(q;\epsilon)$
iii) Proof that every open subset of $\mathbb{R}$  can be written as a countable union of open intervals.
i) I tried to use the triangle inequality but that didn't really help me, I got this at the moment:
Let $n$ be undecided yet, we will try to find an expression for this. Take $ b \in B(x ; 1/n)$, thus $d(b,x) < 1/n$. By the triangle inequality we get $d(a,b) < \epsilon + 1/n$
Intuitively I know that what I have to proof should be true, as a sphere is always open. But the difficulty for me is the fact that $n \in \mathbb{N}$, how can I know for sure that I can always find a proper n.
ii) Again this feels intuitively correct but I feel I need to use question i but I don't know how.
Can anyone help me with some tips on how to get further with the first prove and where to start with the second question?
 A: (i) Let $x \in \mathbb B(a, \epsilon)$. We have to show that there exists a $n \in \mathbb N$, such that for each $y \in \mathbb B\left( x, \frac 1 n \right)$ we have $y \in \mathbb B(a, \epsilon)$. Since $d(x,a) < \epsilon$ you can choose $n \in \mathbb N$ big enough, so that $d(x,a) + \frac 1 n < \epsilon$. By the triangle inequality, we have now for each $y \in \mathbb B\left(x, \frac 1 n \right)$ 
$$ d(y,a) \leq d(y,x) + d(x,a) < \frac 1 n + d(x,a) < \epsilon \; ,$$
i.e. $y \in \mathbb B(a,\epsilon)$.
(ii) Let $a = (a_1, \ldots, a_p) \in \mathbb R^p$. Since $\mathbb Q$ is dense in $\mathbb R$, we find for each $i \in \{1, \ldots, p\}$ a $q_i \in \mathbb Q$, such that $\vert a_i - q_i \vert < \frac{\epsilon}{\sqrt{p}}$. Now calculate the distance $\Vert a - q \Vert$, where $q := (q_1, \ldots, q_p) \in \mathbb Q^p$.
(iii) Let $U \subset \mathbb R$ be an open subset of $\mathbb R$. For each $x \in U$ you find an $\epsilon > 0$, such that $U_x := (x - \epsilon, x + \epsilon) \subset U$. Observe that $$ U = \bigcup_{x \in U} U_x \; .$$ Now think of how you can make this union countable.
A: For i), just choose $n$ such that $\frac1n<\varepsilon-d(a,x)$.
For ii), let $a=(a_1, \ldots, a_p)$. Choose $q=(q_1, \ldots, q_p)\in\mathbb Q^p$ so that $|a_i-q_i|<\frac\varepsilon{p}$, then
$$\begin{align*} d(a,q) &= \left(\sum_{i=1}^p |a_i-q_i|^2\right)^{\frac12}\\ &< \left(\sum_{i=1}^p \left(\frac\varepsilon p\right)^2\right)^{\frac12}\\&\leqslant\sum_{i=1}^p \frac\varepsilon p = \varepsilon.\end{align*}$$
For iii), if $U\subset\mathbb R$ is open, then for each $q\in U\cap\mathbb Q$, let $I_q$ be the union of all open intervals $(a,b)$ with $q\in(a,b)\subset U$. Then $U=\bigcup_{q\in U}I_q$, the countable union of open intervals. For open sets containing rays of the form $(a,\infty)$ or $(-\infty,b)$ you will of course have to include rays in your definition of "open interval".
