I have to show that
(1) Show that $\mathbb{R}^n$ is locally compact.
https://i.stack.imgur.com/qzEAX.png
I have provided a link with my answer since it's long and would take me forever to write in here. Is this proof good enough or do I also need to show that $\mathbb{R}^n$$^-$$^1$ and $\mathbb{R}$ is homeomorphic to $\mathbb{R}^n$?
And as a side question: is there a shorter/easier proof of this?
(2) Assume that X and Y are locally compact Hausdorff spaces. Show that X×Y is locally compact.
Since I proved in (1) that X×Y is locally compact if X and Y are locally compact, do I just refer to that here?
Thank you for your help! Also: I have provided our definition of being locally compact in my link to (1).
In $\mathbb{R}^n$, by Heine-Borel a set is compact iff it is closed and bounded. So any open ball $B(x,\varepsilon) =\{y: d(x,y) < \varepsilon\}$ neighbourhood of $x$ contains a compact neighbourhood of $x$, namely the closed ball $D(x,\delta) = \{y: d(x,y) \le \delta\}$ for $\delta < \varepsilon$ (bounded by definition, closed by a theorem).
(2) is a consequence of the fact that a product of two compact neighbourhoods is a compact neighbourhood in the product.