Yes, so you've proven that [tex]X=\{1/n~\vert~n\in \mathbb{N}\}[/tex] is noncompact, locally compact and Hausdorff. Thus the space has a onepoint compactification. This is the space [tex]\{0\}\cup \{1/n~\vert~n\in \mathbb{N}\}[/tex] with the following topology
[tex]\mathcal{T}=\{A~\vert~X\cap A~\text{open and}~(0\in A~\Rightarrow~X\setminus A~\text{compact})\}[/tex]
This is, by definition, the topology of [tex]\{0\}\cup \{1/n~\vert~n\in \mathbb{N}\}[/tex]. But this space already carries a topology (the Euclidian topology). So you still need to show that these two topologies coincide...
