Consider numbers $x_1 = \sqrt{2}, x_2 = 2\sqrt{2}, ..., x_{n+1} = (n + 1)\sqrt{2}$. Suppose that for each $k = 1, n + 1$ at least one of the numbers $x_k + a_1, x_k + a_2, ..., x_k + a_n$ is rational. As we have $n$ numbers, and try $n + 1$ options, from the Dirichlet principle some two numbers of form $x_i + a_i$ and $x_j + a_j$ will be rational. But then their difference also will be rational, so the number
$$(x_j + a_i) - (x_i + a_j) = x_j - x_i = (j - i)\sqrt{2}$$
will be rational, which isn't true. This contradiction finishes the proof, as for some $x_k = k\sqrt{2}$ all integers $x_k + a_1, x_k + a_2, ..., x_k + a_n$ are irrational.