Dirchlet's Approximation Theorem states that for $k$ real numbers $b\_1, b\_2, \ldots, b\_n$, and $N\in \mathbb{N}$ there exists a $q\in\mathbb{N}$ such that for all $1\le i\le k$, there exists integers $p\_i$ such that $|b\_iq-p\_i|\le \frac{1}{N^{\frac{1}{k}}}$. 


Note that in some sense these approximations can be both lower or upper bounds. For a particular application I need rational approximations of this form in only one direction. In particular, I was curious if the following result holds. For positive real numbers $b\_1,\ldots,b\_k$ and any $\epsilon>0$, there exists a positive integer $q$ such that $0\le (qb\_i-\lfloor qb\_i \rfloor)\le \epsilon$ for $1\le i \le k$. 



"Less than or Equal To" Rational Approximations
This is false in general, consider any irrational $b\_1\in (0,1)$ and $b\_2=1-b\_1$. Then $qb\_1-\lfloor qb\_1 \rfloor+qb\_2-\lfloor qb\_2 \rfloor=1$ for any positive integer $q$, thus if $\epsilon<1/2$, your $q$ does not exist.


On the other hand, if $b\_i$'s and 1 are rationally independent, we may say even more: for any intervals $\Delta\_i\subset (0,1)$ there exist a positive integer $q$ such that fractional part of $qb\_i$ lies in $\Delta\_i$.


