Given two linearly independent vectors $a,b\in\mathbb{R}^2$ we form the lattice $L=\{ma+nb|m,n\in\mathbb{Z}\}$. Now, a proof starts with "choose a nonzero vector in $L$ of smallest length...". Why such a vector exists? In theory the lengths of vectors in $L$ can converge to 0.
Another formulation for the same question with different approach: I am trying to understand the crystallographic restriction theorem (given, e.g., in Armstrong, "Groups and Symmetry", chapter 25). The relevant definition of a crystallographic group is a group of plane isometries whose translation subgroup is generated by two vectors. Hence when this subgroup acts on the origin the resulting orbit is a lattice. Armstrong finds $a,b$ for this lattice by choosing $a$ to be the nonzero vector of smallest length; but again, why do such vector has to exist in the orbit?