Existence of the shortest vector in a lattice

Question

I am studying integer lattices in $\mathbb{R}^n$. I know that since there are no accumulation points in the lattice, the shortest vector always exists. Is there any way that one could prove it?

Answer 1

If there were no shortest vector, then there would be some vector $x$ with $\lVert x \rVert < \varepsilon$ for any $\varepsilon > 0$. So there would be some sequence of nonzero vectors in the lattice with $\lVert x_n \rVert\to 0$. But this is the same as the sequence of the vectors converging to the zero vector in the norm topology on $\mathbb{R}^n$, so then the zero vector would be an accumulation point.