 Solution

The pointwise limit is the constant function $1$. For $0<x<1$, the inequality $1-e^{-u}\le u$ for $u\ge 0$ givesHence the sequence converges uniformly to one on $(0,1)$. Its exact supremum error is $1-e^{-1/n}$, although the supremum is not attained on the open interval. Compactness of the interval is not needed for this uniform convergence bound.