Let $\left\{f_{n}\right\}$ be a pointwise convergent sequence of real-valued functions on a closed interval $[a, b]$. Prove that, if for every $x \in[a, b]$, the sequence $\left\{f_{n}(x)\right\}$ is monotonic in $n$, and if all the functions $f_{n}, n=1,2, \ldots$, and $f=\lim f_{n}$ are continuous, then $f_{n} \rightarrow f$ uniformly on $[a, b]$.

By considering a suitable sequence of functions $\left\{f_{n}\right\}$ on $[0,1)$, show that if the interval is not closed but all other conditions hold, the conclusion of the theorem may fail.