My setting is that $c_1, · · · c_n$ are positive real numbers, and $A = [a_{i,j} ]$ is an $n × n$ non-singular matrix. Assume that $c_1 · · · c_n > | det(A)|.$
I want to prove that the n-linear inequalities
$a_{i,1}x_1 +a_{i,2}x_2 +···+a_{i,n}x_n ≤c_i, 1≤i≤n $
are all satisfied by a nonzero $n-tuple$ $(x_1, · · · , x_n)$ of integers.
We just learned lattice in class, and I know that $det(A)$ is the volume of the fundamental region $F$ of a lattice $\Lambda$, and $(x_1, · · · , x_n)$ here is a basis for $\Lambda$, but I still have no clue how to approach this question. If someone can give me a hint I would really appreciate it. Thanks!