All topologies mentioned are on the cartesian product $\mathbb{R}^{[0,1]}$.
The sequence of functions $(f_k)$, for which, $f_k(x) = x^k, x\in [0,1]$, converges pointwise but not uniformly to $f(x)=\ \begin{cases} 0 & x\in [0,1), \\ 1&x=1\\ \end{cases}$
Therefore, since a sequence of functions $(f_k)$ converges pointwise to $f$ if and only if $(f_k)$ converges to $f$ in the product topology, we have that $f_k \rightarrow f$ in the product topology. Why can't $f$ converge in the box topology? Does that need to be shown every time using the fact that the sequence of functions doesn't converge uniformly or does that follow from the fact that the box topology is finer than the uniform topology?