This follows from the Kroenecker's approximation theorem (With the conditions modified. See edit below).

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.

Edit: As Noam D. Elkies remarked in his answer what we use here is in fact the linear independence over $\mathbb Q$ of the reciprocals of the numbers, $1/a_k$ (Or equivalently in my application of the Kroenecker's approximation theorem, the numbers $a_1/a_k$ ), not the numbers $a_k$ themselves. This means that the question as posed is not true, but it is true when the conditions are modified.

This follows from the Kronecker's approximation theorem (With the conditions modified. See edit below).

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 Kronecker'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.

Edit: As Noam D. Elkies remarked in his answer what we use here is in fact the linear independence over $\mathbb Q$ of the reciprocals of the numbers, $1/a_k$ (Or equivalently in my application of the Kronecker's approximation theorem, the numbers $a_1/a_k$ ), not the numbers $a_k$ themselves. This means that the question as posed is not true, but it is true when the conditions are modified.

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.

This follows from the Kroenecker's approximation theorem (With the conditions modified. See edit below).

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.

Edit: As Noam D. Elkies remarked in his answer what we use here is in fact the linear independence over $\mathbb Q$ of the reciprocals of the numbers, $1/a_k$ (Or equivalently in my application of the Kroenecker's approximation theorem, the numbers $a_1/a_k$ ), not the numbers $a_k$ themselves. This means that the question as posed is not true, but it is true when the conditions are modified.