Define uniform convergence for a sequence $f_{1}, f_{2}, \ldots$ of real-valued functions on the interval $(0,1)$.

For each of the following sequences of functions on $(0,1)$, find the pointwise limit function. Which of these sequences converge uniformly on $(0,1)$ ?

(i) $f_{n}(x)=\log \left(x+\frac{1}{n}\right)$,

(ii) $f_{n}(x)=\cos \left(\frac{x}{n}\right)$.

Justify your answers.