I’m looking at a set of notes on the LLL algorithm, where we have the following setup: let $b_1,\dots,b_n\in\mathbb Z^n$ be a set of $\mathbb R$-linearly independent vectors, and let $\tilde b_1,\dots,\tilde b_n$ be the Gram-Schmidt orhogonalizations of the $b_i$. At one point in one of the proofs (the very end of the proof of Claim 2, in the middle of page 7), it is claimed that for any $j$, $$ |\tilde b_j| \geq \left( |\tilde b_1|^2|\tilde b_2|^2\dots|\tilde b_j|^2\right)^{-1}. $$
Apparently this is obvious, but I’m not quite seeing it. (Well, I can see it for $j=1$ since $\tilde b_1=b_1$ is an integer vector, but I don’t quite see how to extend this to higher $j$.) Does anyone have a proof or a reference for this inequality?