Property of reciprocal lattice

I came across the following property of the reciprocal lattices.

Let be $$\Lambda$$ a Bravais lattice and $$\Lambda^*$$ its reciprocal lattice; let be $$\vec{G} \in \Lambda^*$$ and $$\vec{G}_0$$ the shortest vector in $$\Lambda^*$$ that is parallel to $$\vec{G}$$. Then, the relation between the two vectors is: $$\vec{G}=n\vec{G}_0$$, where n is integer.

I looked for a proof, but I noticed that for example on Ashcroft-Mermin (chapter 5) the proof is left as a problem to the reader. The hint is: suppose the contrary is true (this means "suppose n is not integer", I think) and show that there is a vector parallel to $$\vec{G}$$ that is shorter than $$\vec{G}_0$$. Despite the hint, I'm not able to prove this statement. Does anyone please know this proof?

• How about the following argument: if $n$ is not an integer, then $\mathbf{G}-[n]\mathbf{G}_0=\{n\}\mathbf{G}_0$ is still parallel to $\mathbf{G}$ ($[n]$ is the integer part of $n$, and $\{n\}=n-[n]$), and still belongs to $\Lambda^*$, but with $0<\{n\}<1$, so the length of $\{n\}\mathbf{G}_0$ is smaller than that of $\mathbf{G}_0$. Commented Sep 14, 2022 at 13:41
• You convinced me: if you turn this into an answer I can approve it. Commented Sep 14, 2022 at 13:48

Suppose $$n$$ is not an integer. Denote by $$[n]$$ the integral part of $$n$$, and $$\{n\}=n-[n]$$, so $$0<\{n\}<1$$. Consider $$\vec{G}-[n]\vec{G}_0=\{n\}\vec{G}_0$$, which is still parallel to $$\vec{G}$$, and belongs to $$\Lambda^*$$. But the length of $$\{n\}\vec{G}_0$$ is strictly smaller than $$\vec{G}_0$$, so a contradiction.