euclidean space is a paracompact space I want to show with Heine-Borel that the euclidean space $\Bbb R^n$ is a paracompact space, but I don't know how to start. 
Can anybody help me? Your help will be very appreciated! 
 A: Here is a proof using Heine-Borel Theorem.
Let ${\cal U}$ be a covering of $\Bbb{R}^{n}$. For each integer $m\geq 0$, consider the open ball $B_{m}=\{x\in\Bbb{R}^{n}\colon \|x\|<m\}$ and write
$$
\Bbb{R}^{n}=\bigcup_{m\geq 0}\overline{B_{m}},
$$
where $\overline{B_{m}}$ is the closure of $B_{m}$.
By the Heine-Borel Theorem, we have that each $\overline{B_{m}}$ is compact and hence for each
$m\geq 0$ there exists a finite subset ${\cal U}^{(m)}$
of ${\cal U}$ that covers $\overline{B_{m}}$.
Consider now the covering
$$
{\cal V}:=\bigcup_{m\geq 1}\left\{U\cap(X\setminus\overline{B_{m-1}})\colon
U\in{\cal U}^{(m)}\right\}.
$$
Obviously, ${\cal V}$ is a refinement of ${\cal U}$, as
$U\cap(X\setminus\overline{B_{m-1}})\subseteq U$
for each $U\in{\cal U}^{(m)}$. To see that ${\cal V}$
is a cover of $\Bbb{R}^{n}$, consider a point $x\in\Bbb{R}^{n}$.
There exists a positive integer $m$ such that
$m-1<\|x\|\leq m$, which means that
$x\in\overline{B_{m}}\setminus\overline{B_{m-1}}$.
Thus, $X\setminus\overline{B_{m-1}}$ is an open
neighborhood of $x$ and there exists $U\in{\cal U}^{(m)}$
such that $x\in U$; hence $x\in U\cap (X\setminus\overline{B_{m-1}})$.
Finally, ${\cal V}$ is locally finite indeed. Let
again $x\in\Bbb{R}^{n}$ and consider $k$ such that $\|x\|<k$. Then, $B_{k}$ is an open neighborhood of $x$ that intersects $X\setminus\overline{B_{m-1}}$ only for
$m\leq k$. Therefore, it intersects only finitely many
opens in ${\cal V}$ since each ${\cal U}^{(m)}$ is finite.
A: All metrisable spaces are paracompact, so $\Bbb R^n$ is too (the first theorem pretty hard to prove in full generality though.)
All Lindelöf regular spaces are paracompact so $\Bbb R^n$ is too, this follows from theorem $2$ in this note, e.g. or from theorem $1$ in this one etc. And $\Bbb R^n$ is Lindelöf (from being second countable).
I don't see a particularly large rôle for the Heine-Borel theorem in proving the paracompactness of $\Bbb R^n$.
