I have already posted this question on the mathematics forum, but I suspect the question needs more detailed knowledge than most users have; please excuse the duplicate post. Any help is greatly appreciated.
Preliminaries: As part of Lenstra's algorithm for integer programming (see here, page 4) we compute a linear transformation $\tau$ and a point $z \in \mathbb{R}^n$ which meet certain conditions (step 1). We then compute a reduced LLL-basis $b_1, \ldots, b_n$ for the lattice $\tau \mathbb{Z}^n$ (step 2). In step 3, we need to find a point $v \in \tau \mathbb{Z}^n$ with $$\Vert v - z \Vert \leq \frac{1}{2} \sqrt{n} \max_{i = 1, \ldots, n}{\Vert b_i \Vert}$$ where $\Vert \cdot \Vert$ denotes the euclidian norm.
Claim: The paper I'm working with claims that $$v = \lfloor \tau z \rfloor$$ (where $\lfloor \cdot \rfloor$ is meant component-wise) meets the above condition. However, I cannot prove this claim.
Work so far: The linear transformation $\tau$ can be identified with the matrix $B^{-1} = (b_1 b_2 \ldots b_n)^{-1}$ where the basis vectors $b_i$ are taken as columns of $B$. So we can write \begin{align} \Vert v - z \Vert = \left\Vert \lfloor \tau z \rfloor - z \right\Vert = \left\Vert \sum_{i = 1}^{n}{\left(\lfloor B^{-1}_{i\bullet} z \rfloor - z_i \right)} \right\Vert \enspace. \end{align} This is, however, of limited use (to me) as I do not have any knowledge about the structure of $B^{-1}$. Possibly useful lemmata include Minkowski's inequality, the Cauchy-Schwartz inequality or any properties which follow from $b_1, \ldots, b_n$ being a reduced basis for $\tau \mathbb{Z}^n$.
Thanks in advance for any hints or ideas how to prove the claim.
Further Work: Emil's comment lead me to try if $$v = \tau \left\lfloor \tau^{-1} z \right\rfloor$$ has the required property. Additionally, I replaced the floor function $\lfloor \cdot \rfloor$ through rounding to the next integer $[ \cdot ]$ instead.
This leads to the following estimate: \begin{align} \Vert v - z\Vert &= \Vert \tau \left[ \tau^{-1} z \right] - z \Vert\\ &= \Vert \tau \left[ \tau^{-1} z \right] - \tau \tau^{-1} z \Vert\\ &\leq \Vert \tau \Vert \cdot \Vert \left[ \tau^{-1} z \right] - \tau^{-1} z \Vert\\ &\leq \Vert \tau \Vert \cdot \left\Vert \frac{1}{2} \left( \begin{matrix} 1 \\ \vdots \\ 1\end{matrix} \right)\right\Vert\\ &= \Vert \tau \Vert \cdot \frac{1}{2} \sqrt{n} \end{align}
It remains to be shown that $$ \Vert \tau \Vert \leq \max_{i = 1, \ldots, n}\Vert b_i \Vert $$ holds.
