show that L^+ is non-empty

Question

I want to show that $L^+$ is non-empty where $L$ is a full-rank integer lattice and $L^+$ denotes the set of elements of $L$ having positive coordinates I have an indication that I did not understand $L \otimes \mathbb{Q} ⊆ \mathbb{Z}^n \otimes \mathbb{Q}$ is equal to $\mathbb{Q}^n$; any totally positive element of $\mathbb{Q}^n$ has hence an integer multiple in $L$ thanks

Answer 1

Choose $x\in \mathbb{Q}^n$ with all $x_i > 0$. Since $L$ is a full-rank lattice, there exists some nonzero $n\in \mathbb{Z}$ with $nx\in L$. Assuming without loss of generality that $n > 0$, the point $nx\in L^+$.