This follows from the Kroenecker's approximation theorem.

Assume without loss of generality that $a_k$ are positive, that $c_1 \leq c_k$ and $0 \leq c_k  < a_k$. Let $n_1=n$ and $n_k=\lfloor na_1/a_k\rfloor$ for $k=1,\ldots,K$. Then $n_1 a_1+c_1-(n_k a_k+c_k)=n a_1 -\lfloor na_1/a_k\rfloor a_k+c_1-c_k=c_1-c_k + $  { $ n a_1/a_k $ }$a_k$

By Kroenecker's approximation theorem there exists some $n$ such that
$|c_1-c_k + $ {$n a_1/a_k $}$a_k|<\epsilon/2$ for each $k=1,\ldots,K$. The conclusion follows from the triangle inequality.