The function $f:\mathbb R \to \mathbb R^\infty$ given by $f(t)=(t,t,t,t,t\dots)$ is said to be not continuous in the box topology but continuous in the product topology.

The example given is $U=(-1,1)\times(-1/2,1/2)\times (-1/3,1/3)\times \dots $. Why does this set show the box topology is not continuous by $f$? We can just take the preimage of $U$ and it would just be $U$ correct? I can see that $(-1/n,1/n)$ converges to $0$ which is closed in $\mathbb R$, but for any specific $\mathbb R\_i$ in $\mathbb R^\infty$, we actually do have an open interval even if it is extremely small.