 Solution

For each fixed $x\in (0,1)$, $1/(nx)\to 0$, so the only possible limit is zero. But taking $x_n=1/(2n)$ gives $f_n(x_n)=2$ for every $n$. In fact the supremum of $1/(nx)$ over the interval is infinite for each $n$. Thus the convergence is pointwise but not uniform; uniform convergence cannot be inferred from convergence at each fixed point.