Solution
= Solution
The pointwise limit is the constant function $1$. For $0<x<1$, the inequality $1-e^{-u}\leq u$ for $u\geq0$ gives
$$
|e^{-x/n}-1|=1-e^{-x/n}\leq\frac xn<\frac1n.
$$
Hence \b[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.