Let me share a simple proof I found during a childbirth class 8 years ago:

Let $x_1,\dots,x_d\in\mathbb{R}$ such that $1,x_1,...,x_d$
are linearly independent over $\mathbb{Q}$. Let $\epsilon>0$ and $a_1,\dots,a_d\in\mathbb{R}$ be
arbitrary. We want to show that there are $n\in\mathbb{Z}$ and $y_1,\dots,y_d\in\mathbb{Z}$ such that
$$|nx_i-y_i-a_i|<\epsilon,\quad 1\leq i\leq d.$$
We proceed by induction on $d$, the case of $d=0$ being trivial. The hypothesis is invariant under replacing $x_i$ with $nx_i-y_i$ for any
nonzero $n\in\mathbb{Z}$ and any $y_1,\dots,y_d\in\mathbb{Z}$, while the conclusion only becomes stronger. Hence by Dirichlet's theorem on simultaneous diophantine approximation we can assume from the beginning that
$$|x_i|<\epsilon,\quad 1\leq i\leq d.$$
By the induction hypothesis applied for $x_1/x_d,\dots,x_{d-1}/x_d$,
there are $m\in\mathbb{Z}$ and $y_1,\dots,y_d\in\mathbb{Z}$ such that
such that $r:=(m+a_d)/x_d$ satisfies
$$|rx_i-y_i-a_i|<\epsilon/2,\quad 1\leq i\leq d.$$
Note that for $i=d$ this inequality is automatic with $y_d:=m$. Let $n$ be the closest integer to $r$, then
$$|nx_i-y_i-a_i|\leq |rx_i-y_i-a_i|+|(n-r)x_i|<\epsilon/2+\epsilon/2=\epsilon,\quad 1\leq i\leq d.$$
The proof is complete.

**Remark 1.** I clarified the proof in response to some criticism.

**Remark 2.** Using Dirichlet's theorem again, there are infinitely many $n$'s with the required properties.